A Note on Minimal Separating Function Sets
aa r X i v : . [ m a t h . GN ] S e p A NOTE ON MINIMAL SEPARATING FUNCTION SETS
RAUSHAN BUZYAKOVA AND OLEG OKUNEV
Abstract.
We study point-separating function sets that are minimal withrespect to the property of being separating. We first show that for a compactspace X having a minimal separating function set in C p ( X ) is equivalent tohaving a minimal separating collection of functionally open sets in X . Wealso identify a nice visual property of X that may be responsible for theexistence of a minimal separating function family for X in C p ( X ). We thendiscuss various questions and directions around the topic. Introduction
In this discussion we assume that all spaces are Tychonov. In notations andterminology we will follow [2] and [1] Recall that a function set F ⊂ C p ( X )is point separating if for every distinct x, y ∈ X there exists f ∈ F such that f ( x ) = f ( y ). We will refer to such sets as separating function sets .Ample research has been done on separating function sets. It is natural to lookfor separating sets that have additional nice properties such as compactness,metrizability, etc. Note that it is quite likely that a randomly rendered sep-arating function set contains a proper separating subset with perhaps bettertopological properties. With this in mind we would like to take a closer look atminimal separating function sets. Definition
A set F ⊂ C p ( X ) is a minimal separating function set if F isseparating for X and no proper subset of F is separating. If F is separating,closed (compact), and has not no proper closed (compact) separating subset, F is a minimal closed (compact) separating function set. In this paper we are concerned with the following general problem:
Problem.
What properties of
X, X n are sufficient/necessary for X to have aminimal separating function set in C p ( X ) ? A minimal closed (compact) sepa-rating function set in C p ( X ) ? Mathematics Subject Classification.
Key words and phrases. C p ( X ), minimal point-separating function set, discrete space, thediagonal of a space. In the first part of our study we will concentrate on minimal separating functionssets and then we will zoom into minimal closed (compact) separating functionsets. 2.
Equivalence
In this section we would like to make an observation that for a compact space X having a minimal separating function set in C p ( X ) is equivalent to having aminimal point-separating family of functionally open subsets in X . One part ofthis equivalence is quite obvious but the other requires some work. Recall that afamily of subsets of X is called point-separating if for any two distinct elements x and y in X there exists an element in the family that contains x or y but notboth. Such a family is minimal if no proper subfamily of it is point-separating. Theorem 2.1. If X has a minimal separating family of functionally open setsthen X has a minimal separating function set.Proof. Let O be a family as in the hypothesis. For each O ∈ O , fix f O : X → [0 ,
1] such that f − ((0 , O . Clearly, { f O : O ∈ O} is a minimal separatingfunction set. (cid:3) To reverse the statement of Theorem 2.1, we first prove four very technicalassertions.
Proposition 2.2.
Let X have a τ -sized minimal separating function set. Thenthere exists a τ -sized D ⊂ ( X × X ) \ ∆ X that is discrete and closed in ( X × X ) \ ∆ X .Proof. Fix a minimal separating function set F ⊂ C p ( X ) of cardinality τ . Byminimality, for each f ∈ F we can fix distinct x f , y f in F that are separated by f but not by any other function in F \ { f } . The set D = {h x f , y f i : f ∈ F } isa subset of ( X × X ) \ ∆ X . To show that D is a closed discrete subset in theoff-diagonal, fix h x, y i in X × X with x = y . We need to show that h x, y i canbe separated from D \ {h x, y i} by an open neighborhood.Since F is separating, there exists h ∈ F such that h ( x ) = h ( y ). Let ǫ = | h ( x ) − h ( y ) | /
3. Recall that h does not separate x f from y f if h = f . Therefore,( h ( x ) − ǫ, h ( x ) + ǫ ) × ( h ( y ) − ǫ, h ( y ) + ǫ ) is a desired open neighborhood of h x, y i . (cid:3) In our discussion, we will reference not only the statement of Proposition 2.2but its argument too, namely, the definition of D in terms of elements of F . Note On Minimal Separating Function Sets 3
Lemma 2.3.
There exists a collection { U n : n = 1 , , ... } of open subsets of R with the following properties: (1) { U n : n = 1 , , ... } is point separating, (2) U n contains for each n = 1 , , ... , (3) U n is the only element of the collection that separates from /n .Proof. Put S = { } ∪ { /n : n = 1 , , ... } . Since R is second-countable and S is closed in R , we can fix P = {h A n , B n i : n = 1 , , .. } with the followingproperties:P1: A n , B n are open in R and their closures miss S ,P2: ¯ A n ∩ ¯ B n = ∅ ,P3: For any distinct x, y ∈ R \ S there exists n such that x ∈ A n and y ∈ B n .By P1, for each n , we can fix an open neighborhood O n of S that misses ¯ A n ∪ ¯ B n .Put U n = [ O n \ { /n } ] ∪ A n . Let us show that U = { U n : n = 1 , , ... } is asdesired.Properties (2) and (3) are incorporated in the definition of U n ’s. To show that U is point separating, fix distinct x, y ∈ R . We have three cases:Case ( x, y S ): By P3, there exists n such that x ∈ A n and y ∈ B n . Then x ∈ U n . Since ¯ O n and ¯ A n miss ¯ B n , we conclude that U n misses B n .Therefore, y U n .Case ( x S, y ∈ S ): Pick any positive integer K such that y = 1 /K and x = K . Since x, K S and K = x , by P3, there exists n such that K ∈ A n and x ∈ B n . Then U n contains y but does not contain x .Case ( x, y ∈ S ): We may assume that x = 1 /n . Then U n contains y butnot x .The proof is complete. (cid:3) Lemma 2.4.
There exists a collection { U n : n = 1 , , ... } of open subsets of R with the following properties: (1) { U n : n = 1 , , .. } is point separating, (2) U n is the only element of the collection that separates − /n from /n .Proof. Put S = { } ∪ {± /n : n = 1 , , ... } . Since R is second-countable and S is closed in R , we can fix P = {h A n , B n i : n = 1 , , .. } with the followingproperties:P1: A n , B n are open in R and their closures miss S ,P2: ¯ A n ∩ ¯ B n = ∅ ,P3: For any distinct x, y ∈ R \ S there exists n such that x ∈ A n and y ∈ B n . R. Buzyakova and O. Okunev
By P1, for each n , we can fix an open neighborhood O n of S that misses ¯ A n ∪ ¯ B n .Put U = [ O \ { } ] ∪ A . If n >
1, put U n = [ O n \ { /n, ± / ( n − } ] ∪ A n .Let us show that U = { U n : n = 1 , , ... } is as desired.Property (2) in the conclusion of the lemma’s statement is incorporated in thedefinition of U n ’s. To show that U is point separating, fix distinct x, y ∈ R . If x or y is in R \ S , then the argument is as in Lemma 2.3. We now assume that x, y ∈ S . We have three cases:Case ( x = 0): Then | y | = 1 / ( n −
1) for some n . Hence, U n separates x and y .Case ( x = 1 /n, y = − /n ): Then U n separates x and y .Case ( | x | = 1 /n, | y | = 1 /m, n < m ): Then U m +1 does not contain ± /m = ± y but contains ± /n = ± x .The proof is complete. (cid:3) Note that { U n } n ’s constructed in Lemmas 2.3 and 2.4 are minimal point-separating families of functionally open sets of R . The participating sequencescan be replaced by any non-trivial convergent sequences as well as R can bereplaced by any non-discrete second-countable space.Either of the above two lemmas imply the following: Corollary 2.5.
Any separable metric space X has a minimal separating familyof functionally open sets. We are now ready to prove the reverse of of Theorem 2.1.
Theorem 2.6.
Let X have a minimal separating function set. Then X has aminimal separating collection of functionally open set.Proof. Fix a minimal separating function set F ⊂ C p ( X ). By Corollary 2.5, wemay assume that F is infinite. Partition F into { F α : α < | F |} so that each F α is countably infinite.For each α < | F | we will define U α so that U = ∪{U α : α < | F |} will be aminimal point-separating family of functionally open subsets of X . First, foreach f ∈ F , fix x f and y f in X that are separated by f and no any othermember of F . Construction of U α : Put h α = ∆ { f : f ∈ F α } . Since h α ( X ) is compact, thereexists { f n : n ∈ ω } ⊂ F α and L ∈ h α ( X ) such that lim n →∞ x f n = L . By theargument of Proposition 2.2, the set {h x f , y f i : f ∈ F } is closed and discrete in( X × X ) \ ∆ X . Therefore, lim n →∞ y f n = L too. Since x f = y f for each f we may Note On Minimal Separating Function Sets 5 assume that { x f n : n ∈ ω } ∩ { y f n : n ∈ ω } = ∅ and that no y f n is equal to L .We then have two cases: Case 1. x f n = L for all n ; Case 2. x f n = x f m = L for all n = m .Independently on which case takes place, by Lemmas 2.3 and 2.4, we can finda collection O α = { O nα : n ∈ ω } of open sets in h α ( X ) that separates points of h α ( X ) so that only O nα separates x f n from y f n . Put U α = { h − ( O ) : O ∈ O α } .Let us show that U = ∪{U α : α < | F |} is point separating and minimal withrespect to this property.To show that U is point separating, fix distinct x, y ∈ X . Since F is a pointseparating function set, there exists f ∈ F α ⊂ F that separates x and y . Then h α ( x ) = h α ( y ). Since O α is a point separating family of open subsets of h α ( X ),there exists O ∈ O α that contains exactly one of h α ( x ) and h α ( y ). Hence, h − α ( O ) ∈ U α ⊂ U separates x and y .To show minimality of U with respect to being point separating, fix U ∈ U .Then U = h − ( O ) for some O ∈ O α . Then there exists f ∈ F α such that O is the only element that separates x f and y f . Since f is the only function in F that separates x f and y f , we conclude that U is the only element of U thatseparates x f and y f . (cid:3) Theorems 2.1 and 2.6 imply the promised equivalence.
Theorem 2.7.
A compact space X has a minimal separating function set in C p ( X ) if and only if X has a minimal separating family of functionally opensubsets in X . Minimal Separating Function Sets
In this section the property of having a minimal separating function sets will beoften referred to as the property and will be studies exclusively from the pointof view of C p -theory despite Theorem 2.7. The rational behind this choice isthat any such set is necessarily discrete in itself (Proposition 3.1) and discreteseparating sets have attracted attention of many C p -enthusiasts.First observe that a space can have minimal separating function sets of differentcardinalities. Indeed, one can easily construct a one- and a two-element min-imal separating function families for X = { , , } . While cardinality is not acommon property among minimal function sets, the absence of cluster pointsis. R. Buzyakova and O. Okunev
Proposition 3.1.
Any minimal separating function set is discrete in itself.Proof.
Let F ⊂ C p ( X ) be a minimal separating set. Fix any f ∈ F . Since F isminimal, there exist distinct x, y ∈ X that are separated by f but not by anyother member of F . Put ǫ = | f ( x ) − f ( y ) | /
3. Then U = { g : | g ( x ) − f ( x ) | <ǫ, | g ( y ) − f ( y ) | < ǫ } is an open neighborhood of f that misses F \ { f } . (cid:3) The statement of 3.1 is the main reason we consider our study from the pointof view of C p -theory. In [3] it was proved that if X is a zero-dimensional spaceof pseudoweight τ of uncountable cofinality, then X has a discrete separatingfunction set of size τ if and only if X n has a discrete set of size τ for some n .This and Proposition 2.2 prompt the following question. Question 3.2.
Let X be a zero-dimensional space of (pseudo)weight τ and let ( X × X ) \ ∆ X has a closed discrete subspace of size τ . Does C p ( X ) have aminimal separating set? This observation naturally prompts a question of whether our property is equiv-alent to having a discrete separating function set. We will next identify a naiveexample among well-known spaces that does not have the property under dis-cussion but has a discrete function separating set.
Example 3.3.
The space βω has a discrete separating function set but no min-imal separating function set.Proof. Recall that βω is a zero-dimensional compactum that has a 2 ω -sizedseparating family of clopen sets and a discrete in itself subset of cardinality 2 ω .By [3, Theorem 2.8 and Corollary 2.19], βω has a discrete separating functionset. Since ( βω × βω ) \ ∆ βω is countably compact, it cannot contain a discreteclosed subset. Hence, by Proposition 2.2, βω does not have a minimal separatingfunction set. (cid:3) Now that we have established the existence of spaces without the property, itwould be nice to isolated significant classes of spaces that have the property. Wehave already proved that separable metric spaces have the property (Theorem2.1 and Corollary 2.5). In fact, all metric spaces have the property. We willderive it from a more general statement that we will prove next. For this, by a ( X ) we denote the largest cardinal (if exists) for which there exists a closedsubset A ⊂ X that contains at most one non-isolated point. Recall that w ( X )denotes the weight of X . Note On Minimal Separating Function Sets 7
Theorem 3.4.
Let X be a normal space and a ( X ) = w ( X ) . Then X has aminimal separating function set.Proof. Let S be a closed subset of X with at most one non-isolated point andof size equal to w ( X ). If S has a non-isolated point, denote it by p ∗ . Otherwise,give this name to an arbitrary point of S . Since S is closed, we can fix a w ( X )-sized family A of closed sets with the following properties:A1: For any distinct x, y ∈ X \ S there exist disjoint A x , A y ∈ A containing x and y , respectively.A2: S A = X \ S .Let P be the set of all unordered pairs of disjoint elements of A . Enumerate P as {{ A α , B α } : α < w ( X ) } and enumerate S \ { p ∗ } as { p α : α < w ( X ) } . Nextfor each α < w ( X ), fix a continuous function f α : X → R that has the followingproperties:F1: f α ( S \ { p α } ) = { } ,F2: f α ( p α ) = 1,F3: f α ( A α ) = { / } ,F4: f α ( B α ) = { / } .Such a function exists because A α , B α , S \ { p α } , and { p α } are mutually disjointclosed subsets of a normal space. By A1-A2 and F1-F4, { f α } α is separating.By F1-F2, f α is the only function that separates p ∗ from p α . Hence, { f α } α is aminimal separating family of functions. (cid:3) Corollary 3.5.
Every metric space has a minimal separating function set.
Our next goal is to investigate the behavior of the property within standardstructures and under standard operations. First, we will show that any spacecan be embedded into a space with the property as a closed subset or even asclopen subset. For our next statement, recall that XX ′ is a standard notationfor the Alexandroff double XX ′ of a space X . Proposition 3.6.
The Alexandroff double of any infinite space X has a minimalseparating set of functions of cardinality | X | .Proof. Put τ = | X | . Since X is Tychonoff and | X | = τ , there exists a separatingfamily F ⊂ C p ( X ) that has cardinality τ . Arrange elements of F into a sequence { f α : α < τ } so that every f appears in the sequence infinitely many times. R. Buzyakova and O. Okunev
Enumerate elements of X as { x α : α < τ } . Next, for each α < τ , define g α : XX ′ → R as follows: g α ( p ) = f α ( p ) p ∈ Xf α ( x β ) p = x ′ β f or some β = αf α ( x α ) + 1 p = x ′ α Clearly, g α is continuous and coincides with f α on X . Let us show that G = { g α : α < τ } is a minimal separating family of functions for XX ′ .To prove that G separates points of XX ′ , fix arbitrary distinct p, q ∈ XX ′ .Either both points are in X , or both are in X ′ , or one is in X and the other isin X ′ . Let us consider these three cases separately.Case ( p, q ∈ X ): The conclusion follows from the facts that F separatespoints of X and that g α ’s are extensions of f α ’s.Case ( p, q ∈ X ′ ): There exist distinct α, β such that p = x ′ α , q = x ′ β . Since F separates points of X , there exists γ such that f γ ( x α ) = f γ ( x β ). Since f appears in the enumeration infinitely many times, we may assume that γ
6∈ { α, β } . Then g γ ( p ) = g γ ( q ).Case ( p ∈ X, q ∈ X ′ ): There exist α, β such that p = x α , q = x ′ β . If α = β ,then g α separates p and q . Otherwise, we proceed as in Case 2.Thus, G is point-separating. Finally, only g α separates x α from x ′ α . Therefore, G is minimal. (cid:3) An argument analogous to that of Proposition 3.6 proves the following state-ment.
Proposition 3.7.
Let X be an infinite space and let D X be a discrete spaceof cardinality | X | . Then X ⊕ D X has a minimal separating set of functions ofcardinality | X | . Propositions 3.6 and 3.7 may create an illusion that any kind of doubling guar-antees the presence of the studied property in the resulting ”double-space”.However, the most basic doubling, namely X × { , } need not guarantee anysuch ”improvement”. Indeed, βω × { , } is homeomorphic to itself, and there-fore, does not have the property by Example 3.3. This observation prompts thefollowing. Question 3.8.
Let X × { , } have a minimal separating function set. Does X have such a set? Note On Minimal Separating Function Sets 9
Statements of Example 3.3 and Proposition 3.7 imply that the property is notinherited by closed, open, and even clopen subspaces. The following questionmay still be answered in affirmative.
Question 3.9.
Let X have a minimal separating family of functions. Doesevery open dense subset have the property? Our next observation balances some of the negative statements above.
Theorem 3.10.
Let X α have a minimal separating function set for each α ∈ A .Then, Q α ∈ A X α has a minimal separating function set.Proof. Fix a minimal separating family F α for each X α . Put F = { f ◦ p α : α ∈ A, f ∈ F α } . Clearly, this family is separating (because for any two distinctpoints x , y in the product there is an α ∈ A such that p α ( x ) = p α ( y ), andthere is an f ∈ F α with f ( p α ( x )) = f ( p α ( y )). This family is minimal, becauseif we remove some f ◦ p α , then there are a, b ∈ X α that are not separated by F α \ { f } . Now let x and y be points of the product whose α th coordinates are a and b , and all the remaining coordinates are equal. Then the only projectionthat separates these points is p α , and the images under p α are not separated in X α by F \ { f } , and that’s it. (cid:3) Next, let us look for traces of the property in images and pre-images. Since anydiscrete spaces has the property, the property is not preserved by continuousmaps, even bijective ones. Statements of Example 3.3 and Proposition 3.6 implythat the property is not inherited by taking the inverse image under a continuousinjection. On a positive note, the domain of a continuous bijection has theproperty if the range does. Indeed, let F be a minimal separating function setfor X and let h : Y → X be a continuous bijection. Put G = { f ◦ h : f ∈ F } .Since h is injective, G is a separating family for Y . Since F is minimal, for each f ∈ F we can fix distinct x , x ∈ X that are separated by f but not any otherfunction in F . By surjectivity, y = h − ( x ) , y = h − ( x ) are defined. Clearly, y and y are separated by f ◦ h but not any other function in G . Let us recordour observation as a statement. Proposition 3.11. If X admits a continuous bijection onto a space with aminimal separating set of functions, then X has such a set too. Our discussion prompts the following question.
Question 3.12.
Let X have a minimal separating set of functions and let Y be t -equivalent to X . Does Y have such a set? A Glance at Minimal Separating Closed or Compact FunctionSets
It is always natural to expect more interesting assertions about more rigid struc-tures. With this in mind, we will next apply the minimality requirement withinnarrower classes of function sets. For convenience, let us recall the definitionsfrom the introduction section.
Definition 4.1.
We say that F ⊂ C p ( X ) is a minimal compact separating set ifit is compact and separating and contains no proper compact separating subset.Similarly, F ⊂ C p ( X ) is a minimal closed separating set if it is closed andseparating and contains no proper closed separating subset. Let us start with a non-finite example of a minimal compact separating functionset.
Example 4.2. X = { }∪{ / ( n +1) : n ∈ ω } has a minimal compact separatingset.Proof. Let f n ( x ) = 1 / ( n +1) if x = 1 / ( n +1) and 0 otherwise. Then { f n : n ∈ ω } converges to 0. Therefore, K = { } ∪ { f n : n ∈ ω } is compact. The function f n is the only one that separates 1 / ( n + 1) from 0, and every compact subspace of K that contains all f n is all of K . (cid:3) The argument of Example 4.2 can be used to prove the following statement.
Proposition 4.3.
Let F be a minimal separating function set of X . Then cl C p ( X ) ( F ) is minimal closed separating function set. Statements 4.2 and 4.3 prompt the following questions.
Question 4.4.
Is there a minimal closed (compact) separating function setwithout an isolated point?
Question 4.5.
Does every second-countable (compact) space have a minimalcompact separating functionset? Any Eberlein compactum?
Note On Minimal Separating Function Sets 11
Question 4.6.
Does having a minimal closed (compact) separating function setimply having a minimal separating function set?
Question 4.7.
Is the property of having a minimal compact (closed) functionset productive?
Question 4.8.
Let X have a second countable separating function set. Does X have a minimal separating function set? We would like to finish with two questions that were the main targets of thisstudy and partly addressed by Proposition 2.2 and Theorem 2.7.
Question 4.9.
Is it true that C p ( X ) has a minimal separating τ -sized subset ifand only if X \ ∆ X has a closed discrete τ -sized subset. What if X is compact?What if X is zero-dimensional? Question 4.10.
Is it true that having a minimal function separating set isequivalent to having a minimal separating family of functionally open sets?
References [1] A. Arhangelskii,
Topological Function Spaces , Math. Appl., vol. 78, Kluwer AcademicPublishers, Dordrecht, 1992.[2] R. Engelking,
General Topology , PWN, Warszawa, 1977.[3] R. Buzyakova and O. Okunev,
A Note on Separating Function Sets , Lobachevskii Journalof Mathematics, accepted 2017.
E-mail address : Raushan [email protected]
Miami, Florida, U.S.A.
E-mail address : [email protected]@servidor.unam.mx