Separable reduction theorems by the method of elementary submodels
aa r X i v : . [ m a t h . F A ] M a r SEPARABLE REDUCTION THEOREMS BY THE METHOD OFELEMENTARY SUBMODELS
MAREK C ´UTH
Abstract.
We introduce an interesting method of proving separable reduction theorems -the method of elementary submodels. We are studying whether it is true that a set (function)has given property if and only if it has this property with respect to a special separablesubspace, dependent only on the given set (function). We are interested in properties of sets“to be dense, nowhere dense, meager, residual or porous” and in properties of functions “tobe continuous, semicontinuous or Fr´echet differentiable”. Our method of creating separablesubspaces enables us to combine our results, so we easily get separable reductions of functionproperties such as “be continuous on a dense subset”, “be Fr´echet differentiable on a residualsubset”, etc. Finally, we show some applications of presented separable reduction theoremsand demonstrate that some results of Zaj´ıˇcek, Lindenstrauss and Preiss hold in nonseparablesetting as well. Introduction
The method of elementary submodels is a set-theoretical method which can be used invarious branches of mathematics. A.Dow in [2] illustrated the use of this method in topology,W.Kubi´s in [3] used it in functional analysis, namely to construct projections on Banachspaces. In the present work we slightly simplify and precise the method of elementary sub-models from [3] and we study whether this method can be used to prove separable reductiontheorems which had not been proven by other (more standard) methods.As an success in this way may be considered the following three results. First, we showthat porosity is a separable determined property. Second, we extend the validity of Zaj´ıˇcek’sresult [11; Proposition 3.3] from spaces with separable dual to the general Asplund spaces.And finally, we extend the validity of Preiss’s and Lindenstrauss’s result [6; Theorem 4.8]from spaces c and C ( K ) with a countable compact K to the spaces c (Γ) and C ( K ) with ageneral scattered compact K .It seems that the main advantages of the concept of elementary submodels are: • finite number of results may be combined • the results may be used for more than one space at the same time (having two spaces X and Y which are dependent on each other in some way, we are able to use resultsfor the spaces X , Y and combine them together).Thus, the real strength of this method is revealed when we have proven enough results tocombine them together.The structure of the work is as follows: first we introduce elementary submodels and showsome general results about them. Then we point out how this method is connected with the Mathematics Subject Classification.
Key words and phrases.
Elementary submodel, separable reduction, Fr´echet differentiability, residual set,porous set.The work was supported by the grant SVV-2010-261316. question of separable subspaces. Next, we collect properties of sets and functions which areseparably determined. In the end we produce two extensions of the results contained in [11]and [6] using the method of elementary submodels.Below we recall most relevant notions, definitions and notations.We denote by ω the set of all natural numbers (including 0), by N the set ω \ { } , by R + the interval (0 , ∞ ) and Q + stands for R + ∩ Q . Whenever we say that a set is countable, wemean by this that the set is either finite or infinite and countable. If f is a mapping then wedenote by Rng f the range of f and by Dom f the domain of f . By writing f : X → Y wemean that f is a mapping with Dom f = X and Rng f ⊂ Y . By the symbol f ↾ Z we mark therestriction of the mapping f to the set Z . The closure (resp. interior) of a set A we denoteby A (resp. Int ( A )); the interior relative to a subspace Y we denote by Int Y ( A ).If h X, ρ i is a metric space, we denote by U ( x, r ) the open ball, i.e. the set { y ∈ X : ρ ( x, y ) In this section we introduce the method of creating sets with some special propertiesusing elementary submodels. First we define what those elementary submodels are. Thenwe show which properties they can have. The method discussed in this article is based on aset-theoretical theorem 2.2. It is a combination of the Reflection Theorem and L¨owenheim–Skolem Theorem. We refer reader to Kunen’s book [4], where further details can be found.The idea to use this method in functional analysis comes from the Kubi´s’s article [3]. Someof the following results are therefore based on this article and slightly modified to our situation(namely lemma 2.5 and propositions 2.9, 3.2, 3.4 and 3.5).Let us first recall some definitions:Let N be a fixed set and φ formula. Then relativization of φ to N is a formula φ N whichis a formula obtained from φ by replacing each quantifier of the form “ ∀ x ” by “ ∀ x ∈ N ” andeach quantifier of the form “ ∃ x ” by “ ∃ x ∈ N ”.As an example, if φ := ∀ x ∀ y ∃ z (( x ∈ z ) ∧ ( y ∈ z ))and N = { a, b } , then the relativization of the formula φ to N is φ N = ∀ x ∈ N ∀ y ∈ N ∃ z ∈ N (( x ∈ z ) ∧ ( y ∈ z ))It is clear that φ is satisfied, but φ N is not.If φ ( x , . . . , x n ) is a formula with all free variables shown, then φ is absolute for N if andonly if ∀ a , . . . , a n ∈ N ( φ N ( a , . . . , a n ) ↔ φ ( a , . . . , a n ))A list of formulas, φ , . . . , φ n , is said to be subformula closed if and only if every subformulaof a formula in the list is also contained in the list.Any formula in the set theory can be written using symbols ∈ , = , ∧ , ∨ , ¬ , → , ↔ , ∃ , ( , ) , [ , ]and symbols for variables. Let us assume a subformula closed list of formulas φ , . . . , φ n iswritten in this way. Then it is not difficult to show, that the absoluteness of φ , . . . , φ n for EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 3 N in other words says, that those formulas don’t create any new sets in N . This result iscontained in the following lemma (a proof can be found in [4, Chapter IV Lemma 7.3]): Lemma 2.1. Let N be a set and φ , . . . , φ n subformula closed list of formulas (formulascontaining only symbols ∈ , = , ∧ , ∨ , ¬ , → , ↔ , ∃ , ( , ) , [ , ] and symbols for variables). Then thefollowing are equivalent: (i) φ , . . . , φ n are absolute for N (ii) Whenever φ i is of the form ∃ xφ j ( x, y , . . . y l ) (with all free variables shown), then ∀ y , . . . y l ∈ N [ ∃ x ( φ j ( x, y , . . . y l )) → ( ∃ x ∈ N )( φ j ( x, y , . . . y l ))]The most important result from the set theory for us will be the following theorem (a proofcan be found in [4, Chapter IV Theorem 7.8]). Theorem 2.2. Let φ , . . . , φ n be any formulas and X any set. Then there exists a set M ⊃ X such, that ( φ , . . . , φ n are absolute for M ) ∧ ( | M | ≤ max( ω, | X | ))The set from previous theorem will be often used throughout the paper. Therefore we willuse the following definition: Definition. Let φ , . . . , φ n be any formulas and let X be any countable set. Let M ⊃ X be a countable set satisfying that φ , . . . , φ n are absolute for M . Then we say that M is anelementary submodel for φ , . . . , φ n containing X . We denote this by M ≺ ( φ , ..., φ n ; X ).The relation between X , φ , . . . , φ n and M is often called the elementarity of M .Using lemma 2.1 it is easy to see that the countable union of a monotonne sequence ofelementary submodels is also an elementary submodel. Lemma 2.3. Let ϕ , . . . , ϕ n be a subformula closed list of formulas and let X be any countableset. Let { M k } k ∈ ω be a sequence of sets satisfying (i) M i ⊂ M j , i ≤ j, (ii) ∀ k ∈ ω : M k ≺ ( ϕ , ..., ϕ n ; X ) . Then for M := S k ∈ ω M k it is true, that also M ≺ ( ϕ , ..., ϕ n ; X ) .Proof. It is an easy consequence of lemma 2.1. (cid:3) Let φ ( x , . . . , x n ) be a formula with all free variables shown and let M be some elementarysubmodel for φ . Supposing we want to use the absoluteness of φ for M efficiently, we needto know that a lot of sets are elements of M . The reason is that having a , . . . , a n ∈ M ,the validity of φ ( a , . . . , a n ) and φ M ( a , . . . , a n ) coincides. Therefore, when working withelementary submodels, it is our first aim to force elementary submodel to contain as manyobjects as possible. Let’s see a simple example, how this can be achieved. Example 2.4. Let us have the following formulas: ϕ ( x, a ) := ∀ z ( z ∈ x ↔ (( z ∈ a ) ∨ ( z = a ))) ϕ ( a ) := ∃ xϕ ( x, a )Then for all sets M satisfying M ≺ ( ϕ , ϕ ; ∅ ) it is true that whenever we have a ∈ M , then a ∪ { a } ∈ M . MAREK C ´UTH Proof. Fix a ∈ M . Then ϕ ( a ) is satisfied (the set x satisfying ϕ ( x, a ) is a ∪ { a } ). Fromthe absoluteness of ϕ for M we get, that there exists x ∈ M satisfying ϕ M ( x, a ). Let us fixone such x ∈ M . It is true that ϕ M ( x, a ), and therefore (using absoluteness of ϕ ) ϕ ( x, a )is satisfied as well. But the only possibility how ϕ ( x, a ) can be satisfied is that x = a ∪ { a } .Therefore a ∪ { a } ∈ M . (cid:3) The preceeding example can be generalized to the following lemma: Lemma 2.5. Let φ ( y, x , . . . , x n ) be a formula with all free variables shown and let X be acountable set. Let M be a fixed set, M ≺ ( φ, ∃ yφ ( y, x , . . . , x n ); X ) and let a , . . . , a n ∈ M be such that there exists only one set u satisfying φ ( u, a , . . . , a n ) . Then u ∈ M .Proof. Using the absoluteness of ∃ yφ ( y, x , . . . , x n ) there exists y ∈ M satisfying φ M ( y, a , . . . , a n ).Using the absoluteness of φ we get, that for this y ∈ M the formula φ ( y, a , . . . , a n ) holds.But such y is unique and therefore u = y ∈ M . (cid:3) Using this lemma we can force the elementary submodel M to contain all the needed objectscreated (uniquely) from elements of M . As an example, let us see how it is possible to force M to contain its finite subsets and natural numbers. Proposition 2.6. Let us have the following formulas: ϕ := ∀ z ( z ∈ x ↔ z = z ) ϕ E := ∃ xϕ ( x ) ϕ := ∀ z ( z ∈ x ↔ (( z ∈ u ) ∨ ( z = v ))) ϕ E := ∃ xϕ ( x, u, v ) Then for any nonempty countable set X holds: (i) If M ≺ ( ϕ , ϕ E ; X ) , then ∅ ∈ M . (ii) If M ≺ ( ϕ , ϕ E ; X ) , then for every u, v ∈ M is u ∪ { v } ∈ M . (iii) If M ≺ ( ϕ , ϕ E , ϕ , ϕ E ; X ) , then ω ⊂ M . (iv) If M ≺ ( ϕ , ϕ E , ϕ , ϕ E ; X ) , then for every finite set s ⊂ M is s ∈ M .Proof. ( i ) and ( ii ) follow immediately from the lemma 2.5; ( iii ) follows from ( i ) and ( ii ) byinduction on n ; ( iv ) follows from ( i ) and ( ii ) by induction on the cardinality of s . (cid:3) It would be very laborious and pointless to use only the basic language of the set theory.For example, we often write x < y and we know, that in fact this is a shortcut for a formula ϕ ( x, y, < ) with all free variables shown. In the following text we will use this extendedlanguage of the set theory as we are used to.We will use the following convention. Convention. Whenever we say for a suitable elementary submodel M (the following holds...) ,we mean by this there exists a list of formulas φ , . . . , φ n and a countable set Y such that for every M ≺ ( φ , . . . , φ n ; Y ) (the following holds...) . EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 5 When using this new terminology, we lose the information about formulas φ , . . . , φ n andthe set Y . Anyway, this is not important in applications. Remark . Let us have finite number of sentences T ( a ) , . . . , T n ( a ). Let us assume thatwhenever we fix i ∈ { , . . . , n } , then for a suitable elementary submodel M i the sentence T i ( M i ) is satisfied. Then it is easy to verify, that for a suitable model M the sentence T ( M ) and . . . and T n ( M )is satisfied (it is enough to put together all the lists of formulas and all the sets from thedefinition above).In other words, we are able to combine any finite number of results we have proven using thetechnic of elementary submodels.Let us see some general results about suitable elementary submodels. Proposition 2.8. For a suitable elementary submodel M the following holds:Let f be a function such that f ∈ M . Then (i) Dom f ∈ M (ii) Rng f ∈ M (iii) ( ∀ x ∈ M ∩ Dom f ) ( f ( x ) ∈ M ) Proof. Let us fix an elementary submodel M for formulas marked with ( ∗ ) in the proof belowand all their subformulas. Let f ∈ M be a function. Then Dom f is an object uniquelydefined by the following formula (this formula is the same for all functions f , f is a freevariable in this formula)( ∗ ) ( ∃ D )( ∀ x )( x ∈ D ↔ ( ∃ y : f ( x ) = y )) , and so by the lemma 2.5, Dom f ∈ M . Similarly, Rng f is object uniquely defined by theformula ( ∗ ) ( ∃ R )( ∀ y )( y ∈ R ↔ ( ∃ x : f ( x ) = y )) . By the absoluteness of the formula( ∗ ) ( ∀ x ∈ D ) ( ∃ y : f ( x ) = y )we get that ( iii ) holds. (cid:3) In the sequel we will often start our proofs in the same way. Therefore, by saying “Let usfix a ( ∗ )-elementary submodel M [containing A , . . . , A n ]” we will understand the following:“Let us have formulas ϕ , ϕ E , ϕ , ϕ E from the proposition 2.6 and all the formulas markedwith ( ∗ ) in all the preceeding proofs (and all their subformulas). Add to them formulas markedwith ( ∗ ) in the proof below (and all their subformulas). Denote such a list of formulas by φ , . . . , φ n . Let us fix a countable set X containing the sets ω , Z , Q , Q + , R , R + and allthe common operations and relations on real numbers (+, − , · , :, < ). Fix an elementarysubmodel M for formulas φ , . . . , φ n containing X [such that A , . . . , A n ∈ M ]”.Thus, having such a “( ∗ )-elementary submodel”, we are allowed to use the results of all thepreceeding theorems and propositions.Using this new agreement, let us prove another general proposition. Proposition 2.9. For a suitable elementary submodel M the following holds: MAREK C ´UTH (i) Let S be a finite set. Then S ∈ M ↔ S ⊂ M. (ii) Let S be a countable set. Then S ∈ M → S ⊂ M. (iii) For every natural number n > and for arbitrary ( n + 1) sets a , . . . , a n it is true,that a , . . . , a n ∈ M ↔ h a , . . . , a n i ∈ M. (iv) If A, B ∈ M , then A ∩ B ∈ M , B \ A ∈ M and A ∪ B ∈ M .Proof. Let us fix a ( ∗ )-elementary submodel M . Let us prove that ( ii ) holds. Let S ∈ M bea countable set. If S = ∅ , then S ⊂ M . If S = ∅ , then( ∗ ) ( ∃ f ) ( f is a function from ω onto S ) . Thus, from the elementarity of M , there exists f ∈ M satisfying( f is a function from ω onto S ) M Fix one such function f . Then, using the elementarity of M again, we get that f is a functionfrom ω onto S . Because f is a function with Rng f = S and Dom f = ω ⊂ M , using theproposition 2.8 it is true that S ⊂ M .Let us prove that ( i ) holds. If S ∈ M is finite, then S ⊂ M by ( ii ). If S ⊂ M is finite,then S ∈ M according to the proposition 2.6.( iii ) holds easily from ( i ) by induction on n ∈ ω, n ≥ 1. It is enough to realize, that h a , a i = { a , { a , a }} and h a , . . . , a n i = hh a , . . . , a n − i , a n i .Let us have sets A, B ∈ M . Then, using the lemma 2.5 and the absoluteness of formulas( ∗ ) ( ∃ C )( ∀ x )( x ∈ C ↔ x ∈ A ∧ x ∈ B ) , ( ∗ ) ( ∃ D )( ∀ x )( x ∈ D ↔ x ∈ B ∧ x / ∈ A ) , ( ∗ ) ( ∃ E )( ∀ x )( x ∈ E ↔ x ∈ A ∨ x ∈ B ) , ( iv ) holds. (cid:3) Elementary submodel in the context of normed linear spaces Now we are prepared for some more concrete results concerning mostly metric spacesor normed linear spaces (NLS for short). Before we proceed, let us propose the followingagreements.If h X, ρ i is a metric space (resp. h X, + , · , k · ki is a NLS) and M an elementary submodel,then by saying M contains X (or by writing X ∈ M ) we mean that h X, ρ i ∈ M (resp. h X, + , · , k · ki ∈ M ). If A is a set, then by saying that an elementary model M contains A wemean that A ∈ M .If X is a topological space and M an elementary submodel, then we denote by X M the set X ∩ M . Proposition 3.1. For a suitable elementary submodel M the following holds:Let h X, ρ i be a metric space. Then whenever M contains X , it is true that U ( x, r ) ∈ M whenever x ∈ X ∩ M and r ∈ R + ∩ M . EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 7 Proof. Let us fix a ( ∗ )-elementary submodel M containing X . Let us have x ∈ X ∩ M and r ∈ R + ∩ M . Then U ( x, r ) is an object uniquely determined by the following formula( ∗ ) ( ∃ U )( ∀ z )( z ∈ U ↔ z ∈ X ∧ ρ ( x, z ) < r ) . Thus, according to the lemma 2.5, U ( x, r ) ∈ M . (cid:3) The idea of the following proposition comes from [3]. Proposition 3.2. For a suitable elementary submodel M the following holds:Let X be a NLS. Then whenever M contains X and a set A ⊂ X , it is true that: (i) span( A ) ∩ M is closed separable linear subspace of X . (ii) conv( A ) ∩ M is convex set. (iii) If A is convex, then ( A ∩ M ) = cl w ( A ∩ M ) .In particular, X M is separable subspace of X and X M = cl w ( X ∩ M ) .Proof. Let us fix a ( ∗ )-elementary submodel M containing X and A . Then according to theproposition 2.9, Q ⊂ M and h R , + , − , · , : , < i ∈ M .The elementary submodel M contains functions + : X × X → X and · : R × X → X .Consequently (by the proposition 2.8), X ∩ M is a Q -linear subspace of X . Therefore ( i ) and( ii ) holds. ( iii ) follows easily from ( ii ). (cid:3) Given a Banach space X , list of formulas φ , . . . , φ n and a countable set Y , we are able toget a family of sets M ( X ) := { X M ; M ≺ ( φ , ..., φ n ; Y ) } . By choosing suitable formulas φ , . . . , φ n and suitable set Y , it is possible to force M ( X )to be a family of closed separable subspaces of X having some specific properties. One caneasily join finite number of arguments (lists of formulas) and get another family of separablesubspaces having the same properties as the original family and perhaps even some more.In [7] similar families of closed separable subspaces are used for getting separable reductiontheorems. Those families are called rich. This concept has been originally introduced in [1]by Borwein and Moors. It is possible to find further use of this method for example in [8],where even more references may be found. Definition. Let X be a Banach space. A family R of separable subspaces of X is called rich if (i) for every increasing sequence R i in R , S i ∈ ω R i belongs to R , and(ii) each separable subspace of X is contained in an element of R .The connection between the notion of rich families and elementary submodels is describedin the following lemma. Lemma 3.3. Let X be a Banach space. Then there exists a list of formulas φ , . . . , φ n anda countable set Y such that for every countable set Z and every list of formulas ϕ , . . . , ϕ k such that φ , . . . , φ n , ϕ , . . . , ϕ k is subformulas closed it is true that the family M := { M ; M ≺ ( φ , ..., φ n , ϕ , . . . , ϕ k ; Y ∪ Z ) } satisfies the following conditions: (i) the set { X M ; M ∈ M} is a family of closed separable subspaces of X , MAREK C ´UTH (ii) For every increasing sequence of elementary submodels { M i } i ∈ ω ⊂ M , [ i ∈ ω M i ∈ M and [ i ∈ ω X M i = X S i ∈ ω M i . (iii) For every V separable subspace of X there exists M ∈ M such that V ⊂ X M .Proof. The existence of φ , . . . , φ n and Y such that { X M ; M ∈ M} is a family of closedseparable subspaces follows from the proposition 3.2 above. For ( ii ), let us fix an increasingsequence M i of elementary submodels from the assumption. Then (by the lemma 2.3) itis enough to show that S i ∈ ω X M i = X S i ∈ ω M i . One inclusion follows from the fact that S i ∈ ω X M i ⊂ S i ∈ ω X ∩ M i = X S i ∈ ω M i . The second one holds, because S i ∈ ω X ∩ M i ⊂ S i ∈ ω X ∩ M i = S i ∈ ω X M i . Thus, X S i ∈ ω M i = S i ∈ ω X ∩ M i ⊂ S i ∈ ω X M i . For ( iii ), let ustake any V separable subspace of X and D ⊂ V countable dense set in V . Then taking M ≺ ( φ , ..., φ n , ϕ , . . . , ϕ k ; Y ∪ Z ∪ D ), it is true that V ⊂ X M . (cid:3) In [3] there is introduced a slightly different method of getting the elementary submodels M . It was proven there, that in the case of some classical Banach spaces (namely ℓ p (Γ) and C ( K )) it is possible to describe the subspace X M . Slightly modifying the ideas from [3], weget the same results in our case as well. Definition. Let Γ be a set. Then we denote by suppt Γ the mapping suppt Γ : R Γ → Γ whichmaps x ∈ R Γ to suppt Γ ( x ) = { α ∈ Γ; x ( α ) = 0 } . Proposition 3.4. For a suitable elementary submodel M the following holds:Let X = ℓ p (Γ) , where ≤ p < ∞ and Γ is an arbitrary set. Then whenever M contains X , suppt Γ and Γ , it is true that X M = { x ∈ X ; suppt Γ ( x ) ⊂ M } . Consequently, X M can be identified with the space ℓ p (Γ ∩ M ) .Proof. Let us fix a ( ∗ )-elementary submodel M containing X , suppt Γ , Γ. Let us mark by A the set on the right-hand side. For every x ∈ X ∩ M the set suppt Γ ( x ) is countable and so,according to the propositions 2.8 and 2.9, suppt Γ ( x ) ⊂ M . Thus, x ∈ A . So, X ∩ M ⊂ A ;hence X M ⊂ A . On the other hand, if x ∈ A then arbitrarily close to x we can find y ∈ A such that s = suppt Γ ( y ) ⊂ M is finite and y ( α ) ∈ Q for α ∈ s . Thus, using the proposition2.9, s ∈ M and y ↾ s ∈ M (because y ↾ s = S α ∈ s {h α, y ( α ) i} ). Using the absoluteness of theformula ( ∗ ) ∃ z ∈ X ( z ↾ s = y ↾ s ∧ z ↾ Γ \ s = 0)we get, that y ∈ M . Hence x ∈ X ∩ M = X M . (cid:3) Given a compact space K and an arbitrary elementary submodel M we define the followingequivalence relation ∼ M on K : x ∼ M y ↔ ( ∀ f ∈ C ( K ) ∩ M ) : f ( x ) = f ( y ) . We shall write K/ M instead of K/ ∼ M and we shall denote by q M the canonical quotient map.It is not hard to check that K/ M is a compact Hausdorff space (see [3]).Observe that we can identify the spaces { ϕ ◦ q M : ϕ ∈ C ( K/ M ) } and C ( K/ M ). Indeed,when we define the mapping F ( ϕ ) := ϕ ◦ q M , ϕ ∈ C ( K/ M ) , EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 9 then it is obvious that F is an isometric mapping from C ( K/ M ) onto { ϕ ◦ q M : ϕ ∈ C ( K/ M ) } . Lemma 3.5. For a suitable elementary submodel M the following holds:Let K be a compact space and X = C ( K ) . Let us denote by · the operation of pointwiseproduct of functions in C ( K ) . Then whenever M contains X , · and K , it is true that X M = { ϕ ◦ q M : ϕ ∈ C ( K/ M ) } . Consequently, we can identify X M with the space C ( K/ M ) , where K/ M is a metrizable compactspace.Proof. Let us fix a ( ∗ )-elementary submodel M containing X , · , K . Let us mark by Y theset on the right-hand side. For a given function f ∈ C ( K ) ∩ M we define ϕ ([ x ] M ) := f ( x ) , x ∈ K. It is easy to verify that ϕ is a continuous function. Consequently, f ∈ Y and X M ⊂ Y .For the proof of the second inclusion, let us identify X M with a subspace of C ( K/ M ).Then, according to the propositions 3.2 and 2.8, X M is a closed subspace closed under theoperation · . From the definition of ∼ M it follows that X M separates points in K/ M . Usingthe aboluteness of the formula( ∗ ) ∀ c ∈ R ∃ f ∈ X ( ∀ x ∈ K : f ( x ) = c ) ,M contains every constant rational function; thus, X M contains all the constant functions.From the Stone-Weierstrass theorem we get that X M = C ( K/ M ).Since X M = C ( K/ M ) is a separable space, K/ M is metrizable compact. (cid:3) Properties of sets Let us consider a situation when we have a normed linear space X and we want to recognize,whether a given set A ⊂ X has a property ( P ). For every separable subspace V ⊂ X wewant to find a closed separable subspace V ⊃ V such that A has the property ( P ) in X ifand only if A ∩ V has the property ( P ) in the subspace V .Using the technic of elementary submodels, it is enough to show that for a suitable ele-mentary submodel M (dependent only on the space X and perhaps also on the set A ), theset A has the property ( P ) if and only if A ∩ X M has the property ( P ) in X M .Let us prove the results for properties “to be dense” and “to have empty interior”. Proposition 4.1. For a suitable elementary submodel M the following holds:Let h X, ρ i be a metric space and A, S ⊂ X . Then whenever M contains X , A and S , it istrue that Int S ( A ∩ S ) = ∅ ↔ Int S ∩ X M ( A ∩ S ∩ X M ) = ∅ ,A ∩ S is dense in S ↔ A ∩ S ∩ X M is dense in S ∩ X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A and S . According to theproposition 2.9 we can see that A C ∈ M whenever A ∈ M . Since A is dense in X if and onlyif A C has empty interior in X , it is enough to show the first equivalence.If A ∩ S has nonempty interior in S , then there exists a ball in S , which is a subset of A ∩ S . Thus ( ∗ ) ( ∃ x ∈ S )( ∃ r ∈ R + )( ∀ y ∈ S )( z ∈ U ( x, r ) → z ∈ A ) . In the preceeding formula we use shortcut y ∈ U ( x, r ), which stands for y ∈ X ∧ ρ ( y, x ) < R r .Free variables in the preceeding formula are R + , X, ρ, < R , A, S . Those are contained in M . This allows us to use the elementarity of M . Thus we find x ∈ S ∩ M and r ∈ R + ∩ M suchthat (( ∀ y ∈ S )( z ∈ U ( x, r ) → z ∈ A )) M . Using the elementarity again, U ( x, r ) ∩ S is a subsetof A ∩ S . Consequently, U ( x, r ) ∩ S ∩ X M ⊂ A ∩ S ∩ X M . Since x ∈ U ( x, r ) ∩ S ∩ X M , wehave prooved that A ∩ S ∩ X M contains a nonempty open set in S ∩ X M .Conversely, let us assume that Int S ∩ X M ( A ∩ S ∩ X M ) = ∅ . Then( ∃ x ∈ S ∩ X M )( ∃ r ∈ R + )( U ( x, r ) ∩ S ∩ X M ⊂ A ∩ S ) . Let us take q ∈ (0 , r ) ∩ Q + and x ∈ X ∩ M such that ρ ( x, x ) < q . Then( U ( x , q ) ∩ S ∩ X M ) ⊂ ( U ( x, r ) ∩ S ∩ X M ) ⊂ A ∩ S. For taken x and q holds U ( x , q ) ∩ S ∩ M ⊂ A ∩ S . This can be written as( ∀ y ∈ S ∩ M ) ( ρ ( y, x ) < q → y ∈ A ∩ S ) . Therefore, using the absoluteness of( ∗ ) ( ∀ y ∈ S ) ( ρ ( y, x ) < q → y ∈ A ∩ S ) , we can see that U ( x , q ) ∩ S ⊂ A ∩ S . But the point x is in U ( x , q ) ∩ S . Consequently,Int S ( A ∩ S ) = ∅ . (cid:3) Another set property, which is separably determined, is “to be nowhere dense”. Proposition 4.2. For a suitable elementary submodel M the following holds:Let h X, ρ i be a metric space, G ⊂ X an open set and A ⊂ X . Then whenever M contains X , A and G , it is true that A ∩ G is nowhere dense in G ↔ A ∩ G ∩ X M is nowhere dense in G ∩ X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A and G . According to theproposition 2.9, C ∩ B ∈ M whenever C, B ∈ M . It is well known, that E ⊂ G is nowheredense in G if and only if it is nowhere dense in X (see [5, page 71]). Consequently, it is enoughto prove the proposition for G = X .It is well known, that set A is nowhere dense in a metric space X if and only if the followingformula holds: ∀ x ∈ X ∀ r ∈ R + ∃ y ∈ X ∃ s ∈ R + ( U ( y, s ) ⊂ U ( x, r ) \ A ) . It is easy to check that this is equivalent to the following formula:(1) ( ∗ ) ∀ x ∈ X ∀ r ∈ Q + ∃ y ∈ X ∃ s ∈ Q + ( U ( y, s ) ⊂ U ( x, r ) \ A ) . All the free variables in the preceeding formula are elements of M .Let us prove the implication from the right to the left first. If A is not nowhere dense in X , then ( ∗ ) ∃ x ∈ X ∃ r ∈ Q + ∀ y ∈ X ∀ s ∈ Q + ( U ( y, s ) * U ( x, r ) \ A ) . Using the elementarity of M there exists x ∈ X ∩ M and r ∈ Q + such that:(2) ∀ y ∈ X ∀ s ∈ Q + ( U ( y, s ) * U ( x, r ) \ A ) . Choose an arbitrary y ∈ X M , s ∈ Q + and find such y ∈ X ∩ M that ρ ( y, y ) < s . Then U ( y , s ) ⊂ U ( y, s ). From the validity of (2),( ∗ ) ( ∃ z ∈ X ) ( z ∈ U ( y , s ) \ ( U ( x, r ) \ A )) . EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 11 Using elementarity of M we may fix z ∈ X ∩ M satisfying the formula above. Thus, for given y ∈ X M and s ∈ Q + we have found z ∈ X ∩ M satisfying z ∈ U ( y , s ) \ ( U ( x, r ) \ A ) ⊂ U ( y, s ) \ ( U ( x, r ) ∩ X M \ A ) . Consequently, U ( y, s ) ∩ X M * ( U ( x, r ) ∩ X M ) \ A. The negation of (1) holds in X M ; thus, A ∩ X M is not nowhere dense in X M .For the proof of converse implication, let A be nowhere dense in X . Choose an arbitrary x ∈ X M and r ∈ Q + . Let us find x ∈ X ∩ M satisfying ρ ( x, x ) < r . Then U ( x , r ) ⊂ U ( x, r ). For given point x and number r find y ∈ X and s ∈ Q + from the formula (1).Using elementarity of M we may assume that y ∈ X ∩ M . Consequently, U ( y, s ) ⊂ U ( x , r ) \ A ⊂ U ( x, r ) \ A. The formula (1) is satisfied in X M ; thus, A ∩ X M is nowhere dense in X M . (cid:3) Natural question is, how it is with the property “to be meager”. One implication is simple. Proposition 4.3. For a suitable elementary submodel M the following holds:Let X be a metric space. Then whenever M contains X and a set A ⊂ X , it is true that: A is meager in X → A ∩ X M is meager in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X and A . Let us have a family ofnowhere dense sets { R n } n ∈ ω such that A ⊂ S n ∈ ω R n .Then( ∗ ) ( ∃ ϕ )( ϕ is a function with Dom ϕ = ω, ϕ ( n ) are nowhere dense subsets of X for every n ∈ ω, A ⊂ [ n ∈ ω ϕ ( n )) . From the elementarity of M we may take such a ϕ ∈ M . Consequently (using the propo-sition 2.8), ϕ ( n ) ∈ M for every n ∈ ω .From the proposition 4.2 we get, that for any n ∈ ω the set ϕ ( n ) ∩ X M is nowhere dense in X M . Besides that, A ∩ X M ⊂ S n ∈ ω ( ϕ ( n ) ∩ X M ). Therefore, A ∩ X M is meager in X M . (cid:3) For the converse implication of the preceeding proposition, we need to add some assump-tions. Let us first recall what it means to be somewhere meager. Definition. Let X be a metric space and A ⊂ X . If there are x ∈ X and r > U ( x, r ) ∩ A is meager in X , we say that A is somewhere meager in X .We will need the following easy well-known fact. Lemma 4.4. Let X be a complete metric space and let A ⊂ X have the Baire property. Then X \ A is not meager ↔ A is somewhere meager in X. With the help of the preceeding lemma we are able to prove the converse implication ofproposition 4.3. First, we need to get the result for the properties “Baire property” and “tobe somewhere meager”. Proposition 4.5. For a suitable elementary submodel M the following holds:Let X be a metric space. Then whenever M contains X and a set A ⊂ X , it is true that A is somewhere meager in X → A ∩ X M is somewhere meager in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A and assume that A is some-where meager. With the use of the propositions 2.9 and 3.1, U ( x, r ) ∈ M whenever x ∈ X ∩ M , r ∈ R + ∩ M and C ∩ B ∈ M whenever C, B ∈ M .Because A is somewhere meager, the following formula holds:( ∗ ) ( ∃ x ∈ X )( ∃ r ∈ R + )( U ( x, r ) ∩ A is meager in X ) . From the elementarity of M , we can find x ∈ X ∩ M and r ∈ R + ∩ M such that U ( x, r ) ∩ A is meager in X . Using the proposition 4.3 and the fact that U ( x, r ) ∩ A ∈ M , we can see U ( x, r ) ∩ A ∩ X M is meager in X M . (cid:3) Proposition 4.6. For a suitable elementary submodel M the following holds:Let X be a metric space. Then whenever M contains X and a set A ⊂ X , it is true that: A has Baire property in X → A ∩ X M has Baire property in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A and assume that A has Baireproperty. Then( ∗ ) ( ∃ D )( ∃ P )( D is a G δ subset in X , P is meager subset in X , A = D ∪ P ) . Using the elementarity of M , we may take such D, P ∈ M . With the use of the proposition4.3, P ∩ X M is meager in X M . Consequently, A ∩ X M is union of the G δ set D ∩ X M and themeager set P ∩ X M . (cid:3) Finally, the converse of the proposition 4.3 can be proven under additional assumptions. Theorem 4.7. For a suitable elementary submodel M the following holds:Let X be a complete metric space, G ⊂ X an open set and A ⊂ X set with Baire property.Then whenever M contains X , G and A , it is true that A ∩ G is meager in G ↔ A ∩ G ∩ X M is meager in G ∩ X M ,A ∩ G is residual in G ↔ A ∩ G ∩ X M is residual in G ∩ X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A , G . According to the propo-sition 2.9 it is true, that B ∩ C ∈ M and B C ∈ M whenever B, C ∈ M . It is well known thata set D ⊂ G is meager in X if and only if it is meager in G (see [5, page 83]). Thus, it issufficient to prove the first equivalence for G = X .The implication from the left to the right follows from the proposition 4.3. For the converseimplication, let us assume that A is not meager in X . Consequently, using lemma 4.4, A C is somewhere meager in X . Thus, according to the proposition 4.5, A C ∩ X M is somewheremeager in X M . Then from the propositions 4.6 and 4.4 we get, that A ∩ X M is not meagerin X M . (cid:3) Let us find out, whether the property of sets “to be porous” is separably determined. Whentalking about porosity, we will use the following definition from [9]. Definition. Let X be a metric space, A ⊂ X , x ∈ X and R > 0. Then we define γ ( x, R, A )as the supremum of all r ≥ z ∈ X such that U ( z, r ) ⊂ U ( x, R ) \ A .Further, we define the upper porosity of A at x in the space X as p X ( A, x ) := 2 lim sup R → + γ ( x, R, A ) R EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 13 and the lower porosity of A at x in the space X as p X ( A, x ) := 2 lim inf R → + γ ( x, R, A ) R When it is clear which space X we mean, then we often say upper (lower) porosity of A at x and write p ( A, x ) ( p ( A, x )).We say that A is upper porous ( lower porous , c -upper porous , c -lower porous ) at x if p ( A, x ) > p ( A, x ) > p ( A, x ) ≥ c , p ( A, x ) ≥ c ).We say that A is upper porous ( lower porous , c -upper porous , c -lower porous ) if A is upperporous (lower porous, c -upper porous, c -lower porous) at each y ∈ A . We say that A is σ -upper (lower) porous if it is a countable union of upper (lower) porous sets. Definition. Let h X, ρ i be a metric space and A ⊂ X . Then by d ( · , A ) we understand themapping which maps every x ∈ X to d ( x, A ) := inf { ρ ( x, a ); a ∈ A } .The following lemma is probably well known, but I didn’t find any reference. Lemma 4.8. Let h X, ρ i be a metric space, A ⊂ X and x ∈ A . Let us denote p ( A, x ) := lim sup R → + sup u ∈ U ( x,R ) d( u, A ) R and p ( A, x ) := lim inf R → + sup u ∈ U ( x,R ) d( u, A ) R . Then p ( A, x ) ≤ p ( A, x ) ≤ p ( A, x ) and p ( A, x ) ≤ p ( A, x ) ≤ p ( A, x ) .Proof. In order to show p ( A, x ) ≤ p ( A, x ) and p ( A, x ) ≤ p ( A, x ), it is sufficient to provethat γ ( x, R, A ) ≤ sup u ∈ U ( x,R ) d( u, A ) for every R > 0. Choose some R > r ≥ z ∈ X satisfying U ( z, r ) ⊂ U ( x, R ) \ A . We would like to find u ∈ U ( x, R ) such that r ≤ d( u, A ).But it is easy to check that u = z satisfies those conditions.Now we will prove that p ( A, x ) ≥ p ( A, x ) and p ( A, x ) ≥ p ( A, x ). Take an arbitrary R > u ∈ U ( x, R ) and notice that then d( u, A ) ≤ γ ( x, R, A ).Really, put r = d( u, A ) and z = u . Then for every y ∈ U ( z, r ) is ρ ( u, y ) = ρ ( z, y ) < r = d( u, A ) , so y / ∈ A . Besides that (using the fact that r = d( u, A ) < R , since x ∈ A and u ∈ U ( x, R )), ρ ( y, x ) ≤ ρ ( y, z ) + ρ ( z, x ) < r + R < R. Thus, U ( z, r ) ⊂ U ( x, R ) \ A and d( u, A ) ≤ γ ( x, R, A ).An immediate consequence is, that2 lim sup R → + γ ( x, R, A )2 R ≥ p ( A, x ) , R → + γ ( x, R, A )2 R ≥ p ( A, x ) . Now it is easy to check that also p ( A, x ) ≥ p ( A, x ) and p ( A, x ) ≥ p ( A, x ). (cid:3) The following two propositions show that the first implication about porous sets holds. Proposition 4.9. For a suitable elementary submodel M the following holds:Let h X, ρ i be a metric space. Then whenever M contains X and a set A ⊂ X , it is true that A is not upper porous in X → A ∩ X M is not upper porous in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A . The set A is upper porousin X if and only if the following formula holds: ∀ x ∈ A ∃ m ∈ Q + ∀ R > ∃ R ∈ (0 , R ) γ ( x, R, A ) > Rm. This formula is equivalent to the following one: ∀ x ∈ A ∃ m ∈ Q + ∀ R > ∃ R ∈ (0 , R ) ∃ r > Rm ∃ z ∈ X U ( z, r ) ⊂ U ( x, R ) \ A. Let us notice that this formula is equivalent to the formula, where we take only rationalnumbers R , R and r . It is obvious that we may consider only rational numbers R . Let ustake an arbitrary x ∈ A , m ∈ Q + from the formula above and R ∈ Q + . Then ∃ R ∈ (0 , R ) ∃ r > Rm ∃ z ∈ X U ( z, r ) ⊂ U ( x, R ) \ A. Fix R ∈ (0 , R ), r > Rm and z ∈ X from the formula above. If we take a rational number R q from the interval ( R, min { R , rm } ), then U ( z, r ) ⊂ U ( x, R q ) \ A . Thus, R may be withoutloss of generality considered to be rational. Having now rational number R ∈ (0 , R ), realnumber r > Rm and z ∈ X such that U ( z, r ) ⊂ U ( x, R ) \ A , let us take a rational number r q from the interval ( Rm, r ). Then U ( z, r q ) ⊂ U ( x, R ) \ A . Consequently, the number r may bewithout loss of generality considered to be rational.We have seen that A is not upper porous in X if and only if the following formula holds:( ∗ ) ∃ x ∈ A ∀ m ∈ Q + ∃ R ∈ Q + ∀ R ∈ (0 , R ) ∩ Q + ∀ r ∈ ( Rm, ∞ ) ∩ Q + ∀ z ∈ X U ( z, r ) * U ( x, R ) \ A. (3)Thus, when A is not upper porous in X we are able to find a point x ∈ A from (3). Usingthe elementarity of M , we may assume that x ∈ M . Now fix m ∈ Q + and find R ∈ Q + from the formula (3). Fix R ∈ (0 , R ) ∩ Q + , r ∈ ( Rm, ∞ ) ∩ Q + and z ∈ X M . Then find r ′ ∈ ( Rm, r ) ∩ Q and z ∈ X ∩ M such that ρ ( z, z ) < r − r ′ . Thus, U ( z , r ′ ) ⊂ U ( z, r ). Thenthe following holds: ( ∗ ) ( ∃ y ∈ X ) ( y ∈ U ( z , r ′ ) \ ( U ( x, R ) \ A )) . For r ′ and z we are able to find (using the elementarity of M ) point y ∈ M such that y ∈ U ( z , r ′ ) \ ( U ( x, R ) \ A ) ⊂ U ( z, r ) \ ( U ( x, R ) \ A ) . Consequently, the formula (3) is satisfied in X M and the set A ∩ X M is not upper porousin X M . (cid:3) Proposition 4.10. For a suitable elementary submodel M the following holds:Let X be a metric space. Then whenever M contains X and a set A ⊂ X , it is true that A is not lower porous in X → A ∩ X M is not lower porous in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A . If A is not lower porous,then similarly as in the proof of the proposition 4.9, the following formula holds:( ∗ ) ∃ x ∈ A ∀ m ∈ Q + ∀ R ∈ Q + ∃ R ∈ (0 , R ) ∀ r ∈ ( Rm, ∞ ) ∩ Q + ∀ z ∈ X U ( z, r ) * U ( x, R ) \ A. (4)Using the elementarity of M , let us take x ∈ A ∩ M from the formula above. Then fix m, R ∈ Q + and find R ∈ (0 , R ) such that ∀ r ∈ ( Rm, ∞ ) ∩ Q + ∀ z ∈ X U ( z, r ) * U ( x, R ) \ A. EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 15 Using the elementarity of M we may assume that R ∈ M . Now let us choose an arbitrary r ∈ ( Rm, ∞ ) ∩ Q + and z ∈ X M . Then find r ′ ∈ ( Rm, r ) ∩ Q and z ∈ U ( z, r − r ′ ) ∩ M . Thus, U ( z , r ′ ) ⊂ U ( z, r ). Then the following holds:( ∗ ) ( ∃ y ∈ X ) ( y ∈ U ( z , r ′ ) \ ( U ( x, R ) \ A )) . For r ′ and z we are able to find (using the elementarity of M ) point y ∈ M such that y ∈ U ( z , r ′ ) \ ( U ( x, R ) \ A ). Consequently, X M ∩ U ( z, r ) * U ( x, R ) \ A. Thus, the formula (4) is satisfied in X M and A ∩ X M is not lower porous in X M . (cid:3) To see that the converse implication holds we will follow the ideas presented in [7, page42]. The following result is proven there for a rich family of subspaces (in the case that X is a Banach space). We show the proof for spaces constructed from elementary submodels(which holds even in the case of metric spaces). Lemma 4.11. For a suitable elementary submodel M the following holds:Let h X, ρ i be a metric space and f : X → R a function. Then whenever M contains X , f , itis true that for every R > and x ∈ X M : sup u ∈ U ( x,R ) f ( u ) = sup u ∈ U ( x,R ) ∩ X M f ( u ) . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , f . Fix x ∈ X M and R > u ∈ U ( x,R ) f ( u ) ≤ sup u ∈ U ( x,R ) ∩ X M f ( u ) (the other inequality isobvious). For this purpose, let us take an arbitrary S ∈ Q + satisfying S < sup u ∈ U ( x,R ) f ( u ).Then there exists u ∈ U ( x, R ) such that S < f ( u ). Now, find rational numbers R q , ε ∈ Q + such that R q < R and ρ ( u, x ) < R q − ε . Let us take some x ∈ U ( x, ε ) ∩ M . Then u ∈ U ( x , R q − ε ) and using the absoluteness of the formula( ∗ ) ∃ u ∈ X : ρ ( u, x ) < R q − ε ∧ S < f ( u ) , we get the existence of u ∈ U ( x , R q − ε ) ∩ M ⊂ U ( x, R ) ∩ M such that S < f ( u ). Consequently, S < sup u ∈ U ( x,R ) ∩ X M f ( u ). (cid:3) Proposition 4.12. For a suitable elementary submodel M the following holds:Let X be a metric space. Then whenever M contains X , A ⊂ X and d ( · , A ) , it is true thatfor every x ∈ A ∩ X M A is lower porous at x → A ∩ X M is lower porous at x in the space X M ,A is upper porous at x → A ∩ X M is upper porous at x in the space X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A , d ( · , A ) and fix some x ∈ A ∩ X M such that A is c -upper porous at x for some rational c > 0. Thus, from the lemma4.8 and 4.11 it follows that c ≤ p X ( A, x ) ≤ R → + sup u ∈ U ( x,R ) d( u, A ) R = 2 lim sup R → + sup u ∈ U ( x,R ) ∩ X M d( u, A ) R ≤ R → + sup u ∈ U ( x,R ) ∩ X M d( u, A ∩ X M ) R ≤ p X M ( A ∩ X M , x ) . Consequently, A ∩ X M is c -upper porous in the space X M . The result for lower porosityfollows similarly. (cid:3) Corrolary 4.13. For a suitable elementary submodel M the following holds:Let X be a metric space. Then whenever M contains X , A ⊂ X and d( · , A ) , it is true that A is lower porous in X ↔ A ∩ X M is lower porous in X M ,A is upper porous in X ↔ A ∩ X M is upper porous in X M ,A is σ -lower porous in X → A ∩ X M is σ -lower porous in X M ,A is σ -upper porous in X → A ∩ X M is σ -upper porous in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , A , d ( · , A ). Then the porosityresults follow from the propositions 4.9, 4.10 and 4.12. The σ -porosity results are thenobtained similarly as in the proof of meagerness 4.3 using the absoluteness of the followingtwo formulas( ∗ ) ( ∃ ϕ )( ϕ is function with Dom ϕ = ω, ϕ ( n ) are lower porous subsets of X for every n ∈ ω, A ⊂ [ n ∈ ω ϕ ( n )) . ( ∗ ) ( ∃ ϕ )( ϕ is function with Dom ϕ = ω, ϕ ( n ) are upper porous subsets of X for every n ∈ ω, A ⊂ [ n ∈ ω ϕ ( n )) . (cid:3) It remains unknown to the author whether the converse implication of the preceedingresults about σ -porosity holds as well.5. Properties of functions Let us consider a situation when we have a normed linear space X and we have a function f defined on X . The aim of this section is to say which properties ( P ) of the function f are“separably determined”. To be more concrete, we want to find a closed separable subspace X M such that for every x ∈ X M it is true that: f has the property ( P ) at x ↔ f ↾ X M has the property ( P ) at x. Using the technic of elementary submodels it is possible to combine those results aboutfunctions with the ones about sets.First of the function properties we are interested in is the continuity. Definition. Let h X, ρ i and h Y, σ i be metric spaces, G ⊂ X open subset and f : G → Y afunction. Then we denote by C ( f ) the set of points where f is continuous. Theorem 5.1. For a suitable elementary submodel M the following holds:Let h X, ρ i and h Y, σ i be metric spaces, G ⊂ X open subset and f : G → Y a function. Thenwhenever M contains X , f and Y , it is true that C ( f ) ∈ M and for every x ∈ X M ∩ G : f is continuous at x ↔ f ↾ X M is continuous at x. EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 17 Proof. Let us fix a ( ∗ )-elementary submodel M containing X , Y , f . Then it is true that G ∈ M , since G = Dom( f ). C ( f ) is an object uniquely defined by the formula( ∗ ) ( ∃ C )( ∀ z )( z ∈ C ↔ z ∈ G ∧ f is continuous at z );hence C ( f ) ∈ M . Let us prove the equivalece now. The implication from the left to rightholds for an arbitrary subspace of X . If the function f is not continuous at x ∈ X M ∩ G , thenwe find k ∈ ω such that the following formula holds:(5) ∀ n ∈ ω ∃ y, z ∈ G : (cid:0) y, z ∈ U ( x, n ) ∧ σ ( f ( y ) , f ( z )) > k (cid:1) . Fix n ∈ ω and x ∈ U ( x, n ) ∩ M . Then U ( x , n ) is open set containing x , so there exists l ∈ ω such that U ( x, l ) is a subset of U ( x , n ). According to the formula (5) there are y, z ∈ G satisfying y, z ∈ U ( x, l ) ∧ σ ( f ( y ) , f ( z )) > k . Consequently, the following formula is satisfied:( ∗ ) ∃ y, z ∈ G : (cid:0) y, z ∈ U ( x , n ) ∧ σ ( f ( y ) , f ( z )) > k (cid:1) . All the free variables in this formula are in M , so using the elementarity of M and the factthat U ( x , n ) ⊂ U ( x, n ), we get points y, z ∈ G ∩ M such that(6) y, z ∈ U ( x, n ) ∧ σ ( f ( y ) , f ( z )) > k . We have just shown that for an arbitrary n ∈ ω we are able to find points y, z ∈ G ∩ M satisfying (6). Consequently, the function f ↾ X M is not continuous at x . (cid:3) Having proven that the property ( P ) of the function f (continuity in this case) is separablydetermined, we get that for a set A := { x : f has the property ( P ) at x } the following holds: A ∩ X M = { x : f ↾ X M has the property ( P ) at x } . Combinig this result with the result from the previous section we get the existence of a closedseparable subspace X M such that { x : f has the property ( P ) at x } is dense in X ↔{ x : f ↾ X M has the property ( P ) at x } is dense in X M . Thus, an immediate consequence of the preceeding theorem and results about separablydetermined set properties is the following. Corrolary 5.2. For a suitable elementary submodel M the following holds:Let X and Y be metric spaces, G ⊂ X open subset and f : G → Y a function. Let X becomplete. Then whenever M contains X , Y and f , it is true that C ( f ) is dense in G ↔ C ( f ↾ X M ) is dense in G ∩ X M ,C ( f ) is nowhere dense in G ↔ C ( f ↾ X M ) is nowhere dense in G ∩ X M ,C ( f ) is meager in G ↔ C ( f ↾ X M ) is meager in G ∩ X M ,C ( f ) is residual G ↔ C ( f ↾ X M ) is residual in G ∩ X M ,C ( f ) is not upper porous in X ↔ C ( f ↾ X M ) is not upper porous in X M ,C ( f ) is not lower porous in X ↔ C ( f ↾ X M ) is not lower porous in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , Y , f . Then G ∈ M , because G = Dom( f ). It is well known, that C ( f ) is a G δ set [5, page 207-208]. From the preceedingtheorem, C ( f ) ∩ X M = C ( f ↾ X M ). Therefore we get the wanted result as an immediateconsequence of propositions 4.1, 4.2, 4.9, 4.10 and theorems 4.7, 5.1. (cid:3) Next function property we examine is the lower (upper) semicontinuousity. Let us recallthe definition in metric spaces. Definition. Let X be a metric space, G ⊂ X open subset, f : G → [ −∞ , ∞ ] a function and x ∈ G . If for every sequence { x n } n ∈ ω ⊂ G , x n → x it is true thatlim inf n →∞ f ( x n ) ≥ f ( x ) , then we say that f is lower semicontinuous ( lsc ) at x .If the function ( − f ) is lsc at x , we say that f is upper semicontinuous ( usc ) at x .The following lemma will be used in the proposition saying that the lower (upper) semi-continuity is separably determined property. Lemma 5.3. Let X be a metric space, G ⊂ X open subset, f : G → [ −∞ , ∞ ] a function and x ∈ G . Then f is lsc at x if and only if for every c ∈ Q ∩ ( −∞ , f ( x )) there exists n ∈ ω suchthat f (cid:2) U ( x, n ) ∩ G (cid:3) ⊂ ( c, ∞ ] .Proof. We may assume that f ( x ) > −∞ (if f ( x ) = −∞ , then the lemma is obvious).“ ⇒ ” Suppose that there are number c ∈ Q ∩ ( −∞ , f ( x )) and sequence { x n } n ∈ ω ⊂ G suchthat x n ∈ U ( x, n ), but f ( x n ) ≤ c . Then x n → x , but lim inf n →∞ f ( x n ) ≤ c < f ( x ). Thus, f is not lsc at x .“ ⇐ ” First, let us assume that f ( x ) < ∞ . Fix ε > c ∈ Q ∩ ( f ( x ) − ε, f ( x )) and sequence { x n } n ∈ ω ⊂ G , x n → x . Then there exists k ∈ ω such that f (cid:2) U ( x, k ) ∩ G (cid:3) ⊂ ( c, ∞ ]. Next,there exists n such that for every n ≥ n is x n ∈ U ( x, k ). Then for every n ≥ n is f ( x n ) > c > f ( x ) − ε . Thus, lim inf n →∞ f ( x n ) ≥ f ( x ) − ε . Because we have chosen anarbitrary ε > 0, it is true that lim inf n →∞ f ( x n ) ≥ f ( x ).In the case that f ( x ) = ∞ , we will fix K ∈ ω , c ∈ Q ∩ ( K, ∞ ) and sequence { x n } n ∈ ω ⊂ G , x n → x . Similarly as above it follows that lim inf n →∞ f ( x n ) ≥ K . (cid:3) Proposition 5.4. For a suitable elementary submodel M the following holds:Let X be a metric space, G ⊂ X open subset and f : G → [ −∞ , ∞ ] a function. Thenwhenever M contains X and f , it is true that for every x ∈ X M ∩ G : f is lsc at x ↔ f ↾ X M is lsc at x. Proof. Immediately from the definition it is obvious that the implication from the left to theright holds for any subspace of X . Let us fix a ( ∗ )-elementary submodel M containing X , f and assume that f is not lsc at x ∈ X M ∩ G . Then from the lemma 5.3 we get the existenceof c ∈ Q ∩ ( −∞ , f ( x )) such that for every n ∈ ω exists y ∈ U ( x, n ) ∩ G such that f ( y ) ≤ c .Choose an arbitrary n ∈ ω and find x ∈ U ( x, n ) ∩ M . Then U ( x , n ) ⊂ U ( x, n ) is openset containing x , so there exists l ∈ ω such that U ( x, l ) ⊂ U ( x , n ). For such l ∈ ω thereexists y ∈ U ( x, l ) ∩ G such that f ( y ) ≤ c . Consequently, the following formula holds:( ∗ ) ∃ y ∈ U ( x , n ) ∩ G : f ( y ) ≤ c. EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 19 Using the elementarity of M we find y ∈ U ( x , n ) ∩ G ∩ M ⊂ U ( x, n ) ∩ G ∩ M such that f ( y ) ≤ c . For an arbitrary n ∈ ω we have found y ∈ U ( x, n ) ∩ G ∩ X M such that f ( y ) ≤ c .It follows from the lemma 5.3 that f ↾ X M is not lsc at x . (cid:3) Corrolary 5.5. For a suitable elementary submodel M the following holds:Let X be a metric space, G ⊂ X open subset and f : G → [ −∞ , ∞ ] a function. Let us denoteby − the operation which maps every function h : G → [ −∞ , ∞ ] to the function − h . Thenwhenever M contains X , f and − , it is true that for every x ∈ X M ∩ G : f is usc at x ↔ f ↾ X M is usc at x. Proof. Let us fix a ( ∗ )-elementary submodel M containing X , f , − . Then − f ∈ M , thus itis enough to use the preceeding proposition. (cid:3) The last function property examined in this article is the Fr´echet differentiability. We willuse the following definiton. Definition. Let X and Y be NLS, G ⊂ X open subset, f : G → Y function and x ∈ G .(i) If there exists continuous linear operator A : X → Y such thatlim u → x f ( u ) − f ( x ) − A ( u − x ) k u − x k = 0 , then we say that function f is Fr´echet differentiable at x . The set of the points where f is Fr´echet differentiable we will denote by D ( f ).(ii) For c > ε > δ > D ( f, c, ε, δ ) as the set of all points x ∈ G satisfying (cid:13)(cid:13)(cid:13)(cid:13) f ( y + tv ) − f ( y ) t − f ( y ) − f ( y − hv ) h (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε whenever v ∈ X, k v k = 1 , t > , h > , y ∈ U ( x, δ ) , y − hv ∈ U ( x, δ ) ,y + tv ∈ U ( x, δ ) and min( t, h ) > c k y − x k . The following relationship between sets D ( f, c, ε, δ ) and Fr´echet differentiability is shownin [10]. Lemma 5.6. Let X be NLS, G ⊂ X open subset and let Y be a Banach space. Let f : G → Y be a function. Then f is Fr´echet differentiable at a point x ∈ G if and only if f is continuousat x and x ∈ T n ∈ N S k ∈ N D ( f, n , n , k ) . Using this lemma, it is shown in [10] that the property “to be Fr´echet differentiable” isseparably determined. Let us prove a similar result using the technic of elementary submodels. Theorem 5.7. For a suitable elementary submodel M the following holds:Let X be a NLS, G ⊂ X an open subset and Y a Banach space. Let f : G → Y be a function.Then whenever M contains X , f and Y , it is true that D ( f ) ∈ M and for every x ∈ X M ∩ G : f is Fr´echet differentiable at x ↔ f ↾ X M is Fr´echet differentiable at x. Proof. Let us fix a ( ∗ )-elementary submodel M containing X , Y , f . D ( f ) is an object uniquelydefined by the formula( ∗ ) ( ∃ D )( ∀ z )( z ∈ D ↔ z ∈ D ∧ f is Fr´echet differentiable at z ); hence D ( f ) ∈ M . Fix a point x ∈ X M ∩ G . Then according to the theorem 5.1 it is true that f is continuous at x if and only if f ↾ X M is continuous at x . Thus, using the lemma 5.6, it issufficient to check that x ∈ \ n ∈ N [ k ∈ N D ( f, n , n , k ) ↔ x ∈ \ n ∈ N [ k ∈ N D ( f ↾ X M , n , n , k ) . The implication from the left to the right is obvious (it holds for every subspace of X ).Conversely, let us assume that x / ∈ T n ∈ N S k ∈ N D ( f, n , n , k ). Fix n ∈ N satisfying x / ∈ S k ∈ N D ( f, n , n , k ). Then for every k ∈ ω the following formula holds: ∃ v ∈ X, k v k = 1 , ∃ t, h > , ∃ y ∈ X : y ∈ U ( x, k ) , y − hv ∈ U ( x, k ) , y + tv ∈ U ( x, k ) , min( t, h ) > n ( k y − x k + 0) , (cid:13)(cid:13)(cid:13)(cid:13) f ( y + tv ) − f ( y ) t − f ( y ) − f ( y − hv ) h (cid:13)(cid:13)(cid:13)(cid:13) > n . Mark this formula with ( ∗ ).Let us take some v, t, h and y from the formula above and find η ∈ Q + such that k y − x k < k − η, k y − hv − x k < k − η, k y + tv − x k < k − η, min( t, h ) > n ( k y − x k + 2 η ) . Further, take x ∈ U ( x, η ) ∩ M . Then the following holds: k y − x k ≤ k y − x k + k x − x k < k − η, k y − hv − x k < k − η, k y + tv − x k < k − η, n ( k y − x k + η ) ≤ n ( k y − x k + 2 η ) < min( t, h ) . Using the elementarity of M we get the existence of v ∈ X ∩ M , k v k = 1, t, h ∈ R + ∩ M a y ∈ X ∩ M such that: y ∈ U ( x , k − η ) ⊂ U ( x, k ) , y − hv ∈ U ( x , k − η ) ⊂ U ( x, k ) ,y + tv ∈ U ( x , k − η ) ⊂ U ( x, k ) , min( t, h ) > n ( k y − x k + η ) > n k y − x k , (cid:13)(cid:13)(cid:13)(cid:13) f ( y + kv ) − f ( y ) k − f ( y ) − f ( y − hv ) h (cid:13)(cid:13)(cid:13)(cid:13) > n . Consequently, x / ∈ T n ∈ N S k ∈ N D ( f ↾ X M , n , n , k ). (cid:3) We would like to combine this result with the theorem 4.7, saying that beeing a residualsubset is separable determined property for sets with Baire property in complete metric spaces.The following result comes from [10]. Theorem 5.8. Let X be a normed linear space, G ⊂ X open subset, and let Y be a Banachspace. Let f : G → Y be a function. Then D ( f ) is an F σδ set. Using this result we immediately get the following corrolary (obviously, it is true even more,similarly as in the case of continuity). EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 21 Corrolary 5.9. For a suitable elementary submodel M the following holds:Let X , Y be Banach spaces, G ⊂ X open subset and f : G → Y a function. Then whenever M contains X , Y and f , it is true that D ( f ) is dense in G ↔ D ( f ↾ X M ) is dense in G ∩ X M ,D ( f ) is residual in G ↔ D ( f ↾ X M ) is residual in G ∩ X M . Applications In this last section we show two applications of the theorems proven above. Both of themextend the validity of already known theorems to special nonseparable spaces. In the firstcase we will be interested in the result proven in [11; Proposition 3.3] by Zaj´ıˇcek for spaceswith separable dual. The technic of elementary submodels will allow us to prove that thesame theorem holds in general Asplund spaces. The second application will extend the resultproven in [6; Theorem 4.8] from the spaces C ( K ) with K is a countable compact and fromsubspaces of c to the spaces C ( K ) with K is a general scattered compact and to subspacesof c (Γ) with possibly uncountable set Γ.Separable reductions of the results mentioned above have already been examined using thetechnic of rich families (to remind the concept of rich families, see the section 3). In the firstcase Zaj´ıˇcek in [11; Theorem 5.2] achieved to prove only a weaker variant of the theorem inAsplund spaces. In the second case, the separable reduction to the subspaces of c (Γ) easillyfollows using the work of J.Lindenstrauss, D.Preiss and J.Tiˇser [7; Corrolary 5.6.2] and theresult of Zaj´ıˇcek [11; Theorem 4.7]. However, to the author it is not known whether theextension to spaces C ( K ) with K scattered compact has been proven anywhere else.Let us introduce the first application now.L. Zaj´ıˇcek proved in [11] theorem marked in this text as theorem 6.3. This theorem wasproven for spaces with separable dual. We will show how to use the technic of elementarysubmodels to get the same result for Asplund spaces.In the following, if it is not said otherwise, X will be a Banach space. The equality X = X ⊕ . . . ⊕ X n means that X is the direct sum of non-trivial closed linear subspaces X , . . . , X n and the corresponding projections P i : X → X i are continuous.Recall that X is an Asplund space if each continuous convex real valued function on X isFr´echet differentiable at each point of X except a first category set and that X is Asplundspace if and only if Y ∗ is separable for every separable subspace Y ⊂ X .We will need the following well-known fact (see [11]). Lemma 6.1. Let X be a Banach space, = u ∈ X , and let X = W ⊕ span { u } . Then themapping w ∈ W w + R u ∈ X/ span { u } is a linear homeomorphism. To formulate the result from [11], we need the following definition. Definition. Let f be a real valued function defined on an open subset G of a Banach space X . (i) We say that f is generically Fr´echet differentiable on G if the set D ( f ) of points where f is Fr´echet differentiable is residual in G .(ii) We say that f is strictly differentiable at a ∈ G if there exists x ∗ ∈ X ∗ such thatlim ( x,y ) → ( a,a ) ,x = y f ( y ) − f ( x ) − x ∗ ( y − x ) k y − x k = 0 . (iii) We say that f is essentially smooth ( esm for short) on the line L = a + R v (where a ∈ X , 0 = v ∈ X ) if the function φ ( t ) := f ( a + tv ) is strictly differentiable at a.e.points of its domain. (Obviously, the definition is correct: it does not depend on thechoice of a and v ).(iv) We say that line L is parallel to v (where 0 = v ∈ X ), if there exists a ∈ X such that L = a + R v .(v) We say that f is essentially smooth on a generic line parallel to = v ∈ X , if f isessentially smooth on all lines parallel to v , except a first category set of lines in thefactor space X/ span { v } . Remark . Let X be a NLS, G ⊂ X open subset, f : G → R function, Y a subspace of X and a, v ∈ Y , v = 0. Let us consider the line L = a + R v . Then it follows immediately fromthe definition above that L ⊂ Y and that f is essentially smooth on the line L if and only if f ↾ Y is.The theorem proven in [11; Proposition 3.3] is as follows. Theorem 6.3. Let X = X ⊕ . . . ⊕ X n be a Banach space with a separable dual X ∗ . Let G ⊂ X be an open set and f : G → R a locally Lipschitz function. Let, for each ≤ i ≤ n ,there exists a dense set D i ⊂ S X i such that, for each v ∈ D i , f is essentially smooth on ageneric line parallel to v . Then f is generically Fr´echet differentiable on G . Using the concept of rich families, it is proven in [11; Theorem 5.2] that this result holdswith a slightly stronger assumptions even in the case of nonseparable Asplund spaces. Usingthe technic of elementary submodels we will prove, that the theorem 6.3 holds in exactly thesame form in nonseparable Asplund spaces.For the purpose of proving such a theorem, let us first begin with one lemma. Lemma 6.4. For a suitable elementary submodel M the following holds:Let X be a NLS, X = X ⊕ . . . ⊕ X n . Let P , . . . , P n be the corresponding projections ontosubspaces X , . . . , X n . Then whenever M contains X , P , . . . , P n , it is true that X M = P ( X M ) ⊕ . . . ⊕ P n ( X M ) . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , P , . . . , P n . Then according tothe proposition 2.8 it is true that P i ( X ∩ M ) ⊂ X ∩ M for each i ∈ { , . . . , n } . From thecontinuity of projections P , . . . , P n it follows that P i ( X M ) ⊂ X M for each i ∈ { , . . . , n } .Consequently, X M = P ( X M ) ⊕ . . . ⊕ P n ( X M ). (cid:3) Theorem 6.5. Let X = X ⊕ . . . ⊕ X n be an Asplund space. Let G ⊂ X be an open setand f : G → R a locally Lipschitz function. Let, for each ≤ i ≤ n , there exists a dense set D i ⊂ S X i such that, for each v ∈ D i , f is essentially smooth on a generic line parallel to v .Then f is generically Fr´echet differentiable on G .Proof. Let P , . . . , P n be the continuous projections onto subspaces X , . . . , X n . Accordingto the corrolary 5.9, propositions 4.1, 2.9, 3.2 and lemma 6.4 we know, that there exists a listof formulas ϕ , . . . , ϕ l and a countable set Y such that for the set Z := { X, f, P , . . . , P n , D , . . . , D n , S X , . . . , S X n , Y } and for every elementary submodel M , M ≺ ( ϕ , . . . , ϕ l ; Z ) it is true that:(P1) Every countable set S ∈ M is a subset of M .(P2) X M = P ( X M ) ⊕ . . . ⊕ P n ( X M ). EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 23 (P3) Whenever sets A, S ⊂ X are in M , the following holds: A ∩ S is dense in S ↔ A ∩ S ∩ X M is dense in S ∩ X M . (P4) D ( f ) is residual in G ↔ D ( f ↾ X M ) is residual in G ∩ X M .(P5) X M is separable subspace of X Without loss of generality we may assume that the list of formulas ϕ , . . . , ϕ l is subformulaclosed. Notice, that for every subspace N of X satisfying N = P ( N ) ⊕ . . . ⊕ P n ( N ) it is truethat S X i ∩ N = S P i ( N ) . Really, this equality follows from the fact that S X i ∩ N = S X ∩ X i ∩ N = S X ∩ X i ∩ P i ( N ) = S X ∩ P i ( N ) = S P i ( N ) . Let us define inductively a sequence of elementary submodels { M k } k ∈ ω : • For k = 0 choose an arbitrary elementary submodel M ≺ ( ϕ , ..., ϕ n ; Z ). • Whenever M k is defined, we will find for every i ∈ { , . . . , n } countable subset C k,i of D i ∩ X M k dense in S P i ( X Mk ) = S X i ∩ X M k . Then for every v ∈ C k,i it follows fromthe assumptions and lemma 6.1 that the set { a ∈ G : f is esm on the line a + R v } is residual. Consequently, there exists a G δ dense subset G k,v such that f is esm oneach line parallel to v , intersecting G k,v .Now we let M k +1 to be an elementary submodel for formulas ϕ , . . . , ϕ l containing { Z, C k, , . . . , C k,n , M k , { G k,v } v ∈ S ni =1 C k,i } .Finally, we define M := S k ∈ ω M k . Then according to the lemma 2.3, M ≺ ( ϕ , ..., ϕ n ; Z ).Therefore, ( P − ( P 5) holds for M .We need to verify, that for the space X M and function f ↾ X M the conditions of the theorem6.3 are satisfied. Then according to ( P 4) it is true that f is generically Fr´echet differentiableon G .Since X is an Asplund space, ( X M ) ∗ is separable. Obviously, f ↾ X M is locally Lipschitz.According to ( P X M = P ( X M ) ⊕ . . . ⊕ P n ( X M ). For i ∈ { , . . . , n } we define C i := S k ∈ ω C k,i . Let us verify, that this set is dense in S P i ( X M ) = S X i ∩ X M .Fix an arbitrary ε > y ∈ S X i ∩ X M = S X i ∩ S k ∈ ω ( X ∩ M k ). Then find some y ∈ U ( y, ε ) ∩ S k ∈ ω ( X ∩ M k ) and take k ∈ ω such that y ∈ X ∩ M k . Then y k y k ∈ X M k ∩ S X i .Furthermore, k y k y k − y k ≤ k y k y k − y k + k y − y k = | − k y k| + k y − y k = |k y k − k y k| + k y − y k ≤ k y − y k < ε . Because C k,i is dense in S X i ∩ X M k , there exists c k,i ∈ C k,i ⊂ C i such that k c k,i − y k y k k < ε .Consequently, k c k,i − y k ≤ k c k,i − y k y k k + k y k y k − y k < ε. Notice that thanks to ( P C i ⊂ M for every i ∈ { , . . . , n } . It remains to show that forevery i ∈ { , . . . , n } and v ∈ C i the set R v := { a ∈ G ∩ X M : f ↾ X M is esm on the line a + R v } is residual in X M .Fix an arbitrary v ∈ C i and find k ∈ ω such that v ∈ C k,i . Then R v ⊃ G k,v ∩ X M . Because G k,v ∈ M , we get from ( P 3) that G k,v ∩ X M is dense G δ set in X M . Consequently, R v isresidual in X M . (cid:3) The second application extends validity of the result from [6; Theorem 4.8] marked in thistext as theorem 6.9. This theorem was proven for spaces C ( K ) where K is a countable compactand for subspaces of c . We will show how to use the technic of elementary submodels to getthe same result for spaces C ( K ) where K is a scattered compact and for subspaces of c (Γ)for possibly uncountable set Γ.Recall, that a set A ⊂ T (where T is an arbitrary topological space) is called scattered , ifevery nonempty subset has an isolated point. We wil need the following well-known fact. Lemma 6.6. Let K , L be compact spaces, K scattered, L metrizable and f : K → L contin-uous mapping onto L . Then L is a countable set. Recall that a Banach space Y is said to have the Radon-Nikod´ym property (RNP) if everyLipschitz function f : R → Y is differentiable almost everywhere (or equivalently every such f has a point of differentiability - see [6]).The result of J.Lindenstrauss and D.Preiss uses the notion of Γ-null sets. Therefore, let usgive some basic notations. For further information about this notion see [7, chapter 5].Let X be a Banach space and let T := [0 , N be endowed with the product topology andproduct Lebesgue measure L N . We denote by Γ( X ) the space of continuous mappings γ : T → X having continuous partial derivatives D j γ (we consider one-sided derivatives at points where j -th coordinate is 0 or 1). We equip Γ( X ) with the topology generated by the seminorms k γ k ∞ = sup t ∈ T k γ ( t ) k and k γ k k = sup t ∈ T k D k γ ( t ) k , k ≥ . Equivalently, this topology may be defined by the seminorms k γ k ≤ k = max {k γ k ∞ , k γ k , . . . , k γ k k } . The space Γ( X ) with this topology is a Fr´echet space; in particular it is a Polish spacewhenever X is separable.We define also Γ n ( X ) = C ([0 , n , X ) and consider the norm k · k ≤ n on this space. Notice,that Γ n ( X ) is a subspace of Γ( X ) in the sense that the functions depending on the first n coordinates only are naturally identified with the functions from Γ n ( X ).A Borel subset A ⊂ X is called Γ-null if the set { γ ∈ Γ( X ); L N γ − ( A ) = 0 } is residual inΓ( X ).It comes from [7; Lemma 5.3.2 and Lemma 5.4.1] that the following two lemmas hold. Lemma 6.7. Whenever ( X n ) is an increasing sequence of subspaces of X whose union isdense in X , then S ∞ n =1 Γ n ( X n ) is dense in Γ( X ) . Lemma 6.8. Let A be a Borel subset of a Banach space X .Then the set { γ ∈ Γ( X ); L N γ − ( A ) = 0 } is Borel. The result from [6; Theorem 4.8] comes as follows. Theorem 6.9. The following spaces have the property that every Lipchitz mapping of theminto space with the RNP is Fr´echet differentiable everywhere except a Γ -null set: C ( K ) forcountable compact K , subspaces of c . Let us first focus on the set property “to be Γ-null”. For those purposes we give thefollowing lemmas. EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 25 Lemma 6.10. Let X be a finite dimensional Banach space and let { x , . . . , x n } be basis of X . Then for every k ∈ ω , Γ k ( X ) = { P ni =1 γ i x i ; γ i ∈ Γ k ( R ) } . Proof. For every k ∈ ω , γ ∈ Γ k ( X ) and t ∈ [0 , k there are unique numbers γ ( t ) , . . . , γ n ( t )such that γ ( t ) = P ni =1 γ i ( t ) x i . It is easy to verify that for every i ∈ { , . . . , n } the mapping γ i is an element of Γ k ( R ) and that D j γ ( t ) = P ni =1 D j γ i ( t ) x i whenever j ∈ { , . . . , k } and t ∈ [0 , k . Thus, Γ k ( X ) = { P ni =1 γ i x i ; γ i ∈ Γ k ( R ) } . (cid:3) Lemma 6.11. Let X be a separable Banach space with a countable dense set D . Then Γ( X ) = { P ni =1 γ i x i ; γ i ∈ Γ n ( R ) , x i ∈ D, n ∈ N } . Proof. Let us denote by N either the dimension of X if it is finite, or N = N if X is infi-nite dimensional. Then take a countable linearly dense set { x n } n ∈ N ⊂ D which is linearlyindependent. Denote by X n the subspace span { x i ; i ≤ n } . Then according to the preceedinglemma and lemma 6.7, the set { P ni =1 γ i x i ; γ i ∈ Γ n ( R ) , n ∈ N } is dense in Γ( X ). (cid:3) Remark . The preceeding lemma holds even in the case when X is non-separable (withuncountable set D := X ). This is because the range of every γ ∈ Γ( X ) is separable. Thus,considering that γ ∈ Γ(span { Rng( γ ) } ), we may use the result for separable spaces. Lemma 6.13. For a suitable elementary submodel M the following holds:Let X be a Banach space. Then whenever M contains X and { Γ n ( X ) } ∞ n =1 , it is true that Γ( X ) ∩ M = Γ( X M ) Proof. Let us fix a ( ∗ )-elementary submodel M containing X , { Γ n ( R ) } ∞ n =1 and { Γ n ( X ) } ∞ n =1 (it is not necessary to mention the set { Γ n ( R ) } ∞ n =1 in the assumptions of the lemma as itdoes not depend on the space X - see Convention on the page 4). Then, according to theproposition 2.8, Γ( X ) ∩ M ⊂ Γ( X M ); consequently, Γ( X ) ∩ M ⊂ Γ( X M ).For the other inclusion, denote for every n ∈ N A n := { P ni =1 γ i x i ; γ i ∈ Γ n ( R ) , x i ∈ X ∩ M } . Using the preceeding lemma, it is sufficient to show that for every n ∈ N , A n ⊂ Γ( X ) ∩ M .Let us fix n ∈ N . Using the absoluteness of the formula (for every n ∈ N the formula is thesame - what does change is the free variable Γ n ( R ) in it)( ∗ ) ( ∃ D )( D is countable and dense in Γ n ( R )) , we may find a countable set D ∈ M such that D is dense in Γ n ( R ). Besides that, whenever γ ∈ Γ( R ) ∩ M and x ∈ X ∩ M , than γ x is a function uniquely defined by the formula( ∗ ) ( ∃ f ∈ Γ n ( X ))( ∀ t ∈ [0 , n )( f ( t ) = γ ( t ) x );consequently, γ x ∈ M . As the space Γ( X ) ∩ M is Q -linear, it is true that { P ni =1 γ i x i ; γ i ∈ D, x i ∈ X ∩ M } ⊂ Γ( X ) ∩ M . It is easy to verify that this subset of Γ( X ) ∩ M is dense in A n . (cid:3) Remark . The preceeding lemma is interesting by itself. Observe, that combining it withthe results from previous sections we get, that for every suitable elementary submodel andfor every set A ⊂ Γ( X ) contained in M it is true that A is dense (resp. nowhere dense) inΓ( X ) if and only if A ∩ Γ( X M ) is dense (resp. nowhere meager) in Γ( X M ). When A has the Baire property, then the same equivalence holds for the residuality of A . This result givesus separable subspaces with properties that were not achieved in [7] using the technic of richfamilies (see [7; Lemma 5.6.1]). Corrolary 6.15. For a suitable elementary submodel M the following holds:Let X be a Banach space. Then whenever M contains X , { Γ n ( R ) } ∞ n =1 and a Borel set A , itis true that A is Γ -null in X ↔ A ∩ X M is Γ -null in X M . Proof. Let us fix a ( ∗ )-elementary submodel M containing X , { Γ n ( R ) } ∞ n =1 } and a Borel set A . Then, using the lemma 6.8 and 6.13, { γ ∈ Γ( X ); L N γ − ( A ) = 0 } is residual in Γ( X ) ifand only if { γ ∈ Γ( X M ); L N γ − ( A ∩ X M ) = 0 } is residual in Γ( X M ). (cid:3) Using the preceeding results, we can put forward the promised extension of the theorem6.9. Theorem 6.16. The following spaces have the property that every Lipchitz function of theminto space with the RNP is Fr´echet differentiable everywhere except a Γ -null set: C ( K ) forscattered compact K , subspaces of c (Γ) , where Γ is an arbitrary set.Proof. Let us have a space X from the assumptions (either X = C ( K ) for scattered compact K , or X ⊂ c (Γ)), a Banach space Y with RNP and a Lipschitz function f : X → Y .Using the preceeding corrolary 6.15 and the theorem 5.7, choose an elementary submodel M satisfying: • X M is a separable subspace of X • f is Fr´echet differentiable everywhere except a Γ-null set in X if and only if f ↾ X M isFr´echet differentiable everywhere except a Γ-null set in X M If X = C ( K ), then choose (using lemma 3.5) such an elementary submodel M , that inaddition it holds that X M = C ( K/ M ), where K/ M is metrizable compact and a continuousimage of K . From the lemma 6.6 it follows that K/ M is a countable compact. Then from thetheorem 6.9 it follows that f ↾ X M is Fr´echet differentiable everywhere except a Γ-null set in X M , so f is Fr´echet differentiable everywhere except a Γ-null set.If X = c (Γ), then X M is a separable subspace of X , so X M is a subspace of c . Then,using the same arguments as above, f is Fr´echet differentiable everywhere except a Γ-nullset. (cid:3) Acknowledgments. The author would like to thank Ondˇrej Kalenda for suggesting the topicand for many useful remarks. My gratitude belongs also to J. Tiˇser for useful remark whichhelped to prove the lemma 6.13. References [1] J.M.Borwein, W.Moors: Separable determination of integrability and minimality of the Clarke subdiffer-ential mapping , Proc. Amer. Math. Soc. (1999), 215-221[2] A.Dow: An introduction to applications of elementary submodels to topology , Topology Proc. (1988),17-72[3] W.Kubi´s: Banach spaces with projectional skeletons , J. Math. Anal. Appl. (2009), no. 2, 758-776.[4] K.Kunen: Set Theory , Stud. Logic Found. Math., vol. 102, North-Holland Publishing Co., Amsterdam,1983.[5] K.Kuratowski: Topology , vol. 1, Academic Press, New York, 1966.[6] J.Lindenstrauss, D.Preiss: On Fr´echet differentiability of Lipschitz maps between Banach spaces , Annalsof Math. (2003), 257288. EPARABLE REDUCTION THEOREMS BY THE METHOD OF ELEMENTARY SUBMODELS 27 [7] J.Lindenstrauss, D.Preiss, J. Tiˇser: Fr´echet differentiability of Lipschitz functions and porous sets inBanach spaces , monograph in preparation[8] W.Moors, J.Spurn´y: On the topology of pointwise convergence on the boundaries of L -preduals , Proc.Amer. Math. Soc. ( ) (2009), 1421-1429.[9] L.Zaj´ıˇcek: On sigma-porous sets in abstract spaces , Abstract and Appl. Analysis , 509-534.[10] L.Zaj´ıˇcek: Fr´echet differentiability, strict differentiability and subdifferentiability , Czechoslovak Math. J. ( ) (1991), 471-489.[11] L.Zaj´ıˇcek: Generic Fr´echet differentiability on Asplund spaces via a.e. strict differentiability on manylines , preprint E-mail address : [email protected]@karlin.mff.cuni.cz