On scatteredly continuous maps between topological spaces
aa r X i v : . [ m a t h . G T ] M a r ON SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES
TARAS BANAKH AND BOGDAN BOKALO
Abstract.
A map f : X → Y between topological spaces is defined to be scatteredly continuous iffor each subspace A ⊂ X the restriction f | A has a point of continuity. We show that for a function f : X → Y from a perfectly paracompact hereditarily Baire Preiss-Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset A ⊂ X theset D ( f | A ) of discontinuity points of f | A is nowhere dense in A ), (ii) the piecewise continuity ( X canbe written as a countable union of closed subsets on which f is continuous), (iii) the G δ -measurability(the preimage of each open set is of type G δ ). Also under Martin Axiom, we construct a G δ -measurablemap f : X → Y between metrizable separable spaces, which is not piecewise continuous. This answersan old question of V.Vinokurov. Introduction
In this paper we study scatteredly continuous maps of topological spaces and show that under somerestrictions on the domain and range the scattered continuity is equivalent to many other function prop-erties considered in the literature (such as: piecewise continuity, G δ -measurability, weak discontinuity,etc.).By definition, a map f : X → Y between topological spaces is scatteredly continuous if for each non-empty subspace A ⊂ X the restriction f | A has a point of continuity. Such maps were introduced in [AB]and appear naturally in analysis, see [BKMM].By its spirit the definition of a scatteredly continuous map resembles the classical definition of apointwise discontinuous map due to R.Baire [B]. We recall that a map f : X → Y is called pointwisediscontinuous if for each non-empty closed subspace A ⊂ X the restriction f | A has a continuity point.Pointwise discontinuous maps can be characterized in many different ways. In particular, the followingclassical theorem of R.Baire is well-known: Theorem 1.1 (Baire) . For a real-valued function f : X → R from a complete metric space X thefollowing conditions are equivalent: (1) f is pointwise discontinuous; (2) f is F σ -measurable (in the sense that for any open set U ⊂ Y the preimage f − ( U ) is of type F σ in X );(3) f is of the first Baire class (i.e., f is a pointwise limit of a sequence of continuous functions).A similar characterization holds also for scatteredly continuous maps: Theorem 1.2.
For a real-valued function f : X → R from a complete metric space X the followingconditions are equivalent: (1) f is scatteredly continuous; (2) f is weakly discontinuous (which means that for every non-empty set A ⊂ X the set D ( f | A ) ofdiscontinuity points of f | A is nowhere dense in A );(3) f is piecewise continuous (which means that X has a countable closed cover C such that f | C iscontinuous for every C ∈ C );(4) f is G δ -measurable (which means that for every open set U ⊂ R the preimage f − ( U ) is of type G δ in X ); Mathematics Subject Classification.
Key words and phrases.
Scatteredly continuous map, weakly discontinuous map, piecewise continuous map, G δ -measurable map, Preiss-Simon space.Dedicated to Tsugunori Nogura on the occasion of his 60th birthday. (5) f is of the stable first Baire class (which means that there is a sequence ( f n : X → R ) n ∈ ω ofcontinuous functions that stably converges to f in the sense that for every x ∈ X there is n ∈ ω such that f m ( x ) = f ( x ) for all m ≥ n ).This characterization theorem is a combined result of investigations of many authors, in particular[BM], [Kir], [CL ], [Sol], and holds in a bit more general setting. Unfortunately all known generalizationsof Theorem 1.2 concern maps with metrizable range or domain. In this paper we treat the general non-metrizable case. Our main result is Theorem 8.1 which unifies the results of sections 4–7 and says that theconditions (1)–(4) of Theorem 1.2 are equivalent for any map f : X → Y from a perfectly paracompacthereditarily Baire Preiss-Simon space X to a regular space Y . If, in addition, Y ∈ σ AE( X ), then allthe five conditions of Theorem 1.2 are equivalent. In Section 9 we present some examples showing thenecessity of the assumptions in Theorem 8.1. In particular, we answer an old question of V.Vinokurov[Vino] constructing a G δ -measurable map f : X → Y between separable metrizable spaces that fails tobe piecewise continuous.1.1. Terminology and notations.
Our terminology and notation are standard and follow the mono-graphs [Ar], [En]. A “space” always means a “topological space”. Maps between topological spaces canbe discontinuous.For a subset A of a topological space X by cl X ( A ) or A we denote the closure of A in X while Int( A )stands for the interior of A in X . For a function f : X → Y between topological spaces by C ( f ) and D ( f ) = X \ C ( f ) we denote the sets of continuity and discontinuity points of f , respectively.By R and Q we denote the spaces of real and rational numbers, respectively; ω stands for the space offinite ordinals (= non-negative integers) endowed with the discrete topology. We shall identify cardinalswith the smallest ordinals of the given size.By ω <ω we denote the set of all finite sequences s = ( n , . . . , n k ) of non-negative integer numbers.For such a sequence s = ( n , . . . , n k ) and a number n ∈ ω by s ˆ n = ( n , . . . , n k , n ) we denote theconcatenation of s and n .2. Some elementary properties of scatteredly continuous maps
We start recalling the definition of a scatteredly continuous map, the principal concept of this paper.Then we shall prove some elementary properties of such maps and present some counterexamples.
Definition 2.1.
A map f : X → Y between topological spaces is called scatteredly continuous if for anynon-empty subspace A ⊂ X the restriction f | A : A → Y has a point of continuity.Scatteredly continuous maps can be considered as mapping counterparts of scattered topological spaces.Let us recall that a topological space X is scattered if each non-empty subspace A of X has an isolatedpoint. It is clear that a topological space X is scattered if and only if any bijective map f : X → Y to adiscrete space Y is scatteredly continuous. In fact, a bit more general result is true. Proposition 2.2.
A space X is scattered if and only if any bijective map f : X → Y to a scattered space Y is scatteredly continuous.Proof. The “if” part is trivial. To prove the “only if” part, assume that f : X → Y is a scatteredlycontinuous map to a scattered space Y . We need to prove that each non-empty subspace A ⊂ X has anisolated point. Since f is scatteredly continuous, the set C ( f | A ) of continuity points of the restriction f | A is dense in A . The image f ( C ( f | A )) ⊂ Y , being scattered, contains an isolated point y . By thecontinuity of f | C ( f | A ), the point x = ( f | A ) − ( y ) is isolated in C ( f | A ) and by the density of C ( f | A )in A , x is isolated in A . (cid:3) The scattered continuity of maps f : X → Y defined on spaces of countable tightness can be detectedon countable subspaces of X . We recall that a topological space X has countable tightness if for everysubset A ⊂ X and any point a ∈ A there is a countable subset B ⊂ A with a ∈ B . Proposition 2.3.
A map f : X → Y defined of a space X of countable tightness is scatteredly continuousif and only if for every countable subspace Q ⊂ X the restriction f | Q has a continuity point. N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 3
Proof.
The “only if” part of this proposition is trivial. To prove the “if” part, assume that f : X → Y is not scatteredly continuous and X has countable tightness. Since f is not scatteredly continuous, thereis a subset D ⊂ X such that f | D has no continuity point.By induction on the tree ω <ω we can construct a sequence ( x s ) s ∈ ω <ω of points of D such that for everysequence s ∈ ω <ω the following conditions hold: • x s is a cluster point of the set { x s ˆ n : n ∈ ω } ; • f ( x s ) is not a cluster point of { f ( x s ˆ n ) : n ∈ ω } .We start the inductive construction taking any point x ∅ ∈ D . Assuming that for some finite sequence s ∈ ω <ω the point x s ∈ D has been chosen, use the discontinuity of f | D at x s to find a neighborhood U of f ( x s ) such that x s is a cluster point of D \ f − ( U ). Since X has countable tightness there is acountable set { x s ˆ n : n ∈ ω } ⊂ D \ f − ( U ) whose closure contains the point x s . This completes theinductive construction.Then the set Q = { x s : s ∈ ω <ω } is countable and f | Q has no continuity point. (cid:3) We finish this section with an example of two scatteredly continuous maps whose composition is notscatteredly continuous.
Example 2.4.
Let f : R → R Q be the identity map from the real line equipped with the standardtopology τ to the real line endowed with the topology generated by the subbase τ ∪ { Q } . Also let χ Q : R Q → { , } be the characteristic function of the set Q . It is easy to show that the maps f : R → R Q and χ Q : R Q → { , } are scatteredly continuous while their composition χ Q ◦ f : R → { , } is everywherediscontinuous (and hence fails to be scatteredly continuous).3. The index of scattered continuity
The scattered continuity of a map f : X → Y can be measured by an ordinal index sc( f ) defined asfollows. Consider the decreasing transfinite sequence ( D α ( f )) α of subspaces of X defined by recursion: D ( f ) = X and D α ( f ) = T β<α D ( f | D β ( f )) for an ordinal α >
0. The transfinite sequence ( D α ( f )) α willbe called the discontinuity series of f . The scattered continuity of f implies that D α ( f ) = ∅ for someordinal α . The smallest ordinal α with D α ( f ) = ∅ is called the index of scattered continuity of f and isdenoted by sc( f ).A decreasing transfinite sequence ( X α ) α<β of subsets of a set X will be called a vanishing series if X = X , X β = ∅ , X α +1 = X α for all α < β , and X α = T γ<α X γ for each limit ordinal α ≤ β . A typicalexample of a vanishing series is the discontinuity series ( D α ( f )) α ≤ sc( f ) of a scatteredly continuous map f . Proposition 3.1.
A map f : X → Y is scatteredly continuous if and only if there is a vanishing series ( X α ) α ≤ β of subsets of X such that X α +1 ⊃ D ( f | X α ) for every ordinal α < β . The smallest length β ofsuch a series is equal to sc( f ) , the index of scattered continuity of f .Proof. The “only if” part follows from the definition of the discontinuity series ( D α ( f )) of f . To provethe “if” part, assume that ( X α ) α ≤ β is a vanishing sequence of subsets of X such that X α +1 ⊃ D ( f | X α )for every ordinal α < β . To show that f is scatteredly continuous, take any non-empty subset A ⊂ X and find the smallest ordinal α ≤ β such that A X α . We claim that α is not a limit ordinal. Otherwise A X α = T γ<α X γ would imply that A X γ for some γ < α , which would contradict the definition of α . So α = γ + 1 for some ordinal γ . Take any point a ∈ A \ X γ +1 . It follows from the definition of α that A ⊂ X γ . Since f | X γ is continuous at each point of the set X γ \ X γ +1 , the restriction f | A is continuousat a . This proves the scattered continuity of f .Next, we show that sc( f ) ≤ β . This will follow as soon as we prove that D α ( f ) ⊂ X α for all ordinals α ≤ β . This can be done by induction. For α = 0 we get D ( f ) = X = X .Assume that for some ordinal α ≤ β we have proved that D γ ( f ) ⊂ X γ for all γ < α . We should provethat D α ( f ) ⊂ X α . If α is a limit ordinal, then D α ( f ) = \ γ<α D ( f | D γ ( f )) = \ γ<α D γ ( f ) ⊂ \ γ<α X γ = X α TARAS BANAKH AND BOGDAN BOKALO by the inductive hypothesis. If α = γ + 1 is a successor ordinal, then the inclusion X γ +1 ⊃ D ( f | X γ )implies that f | X γ is continuous at points of X γ \ X γ +1 . By the inductive hypothesis, D γ ( f ) ⊂ X γ andhence f | D γ ( f ) is continuous at points of D γ ( f ) \ X γ +1 . Consequently, D α ( f ) = D ( f | D γ ( f )) ⊂ X γ +1 = X α . (cid:3) The discontinuity series will help us to characterize scatteredly continuous maps via well-orders.
Proposition 3.2.
A map f : X → Y between topological spaces is scatteredly continuous if and onlyif there exists a well-order ≺ on the set X such that for any non-empty subset A ⊂ X the restriction f | A : A → Y is continuous at the point x = min( A, ≺ ) .Proof. The “if” part is trivial. To prove the “only if” part, take any scatteredly continuous map f : X → Y and consider the discontinuity series ( D α ( f )) α< sc( f ) of f . On each set D α ( f ) \ D α +1 ( f ) fix awell-order ≺ α . The well-orders ≺ α , α < sc( f ), compose a well-order ≺ on X defined by x ≺ y if either x, y ∈ D α ( f ) \ D α +1 ( f ) for some α and x ≺ α y or else x ∈ D α ( f ) and y / ∈ D α ( f ) for some α .Given any non-empty subset A ⊂ X we should check that f | A is continuous at the point a = min( A, ≺ ).Find an ordinal α such that a ∈ D α ( f ) \ D α +1 ( f ) and observe that A ⊂ D α ( f ). Since D α +1 ( f ) = D ( f | D α ( f )), a is a continuity point of f | D α ( f ) and hence of f | A . (cid:3) Scatteredly continuous and weakly discontinuous maps
The pathology described in Example 2.4 cannot happen in the realm of regular spaces: in this sectionwe shall show that the composition of scatteredly continuous maps between regular spaces is scatteredlycontinuous. For this we first consider compositions of scatteredly continuous and weakly discontinuousmaps.Following [Vino], we define a map f : X → Y to be weakly discontinuous if for every subspace Z ⊂ X the set D ( f | Z ) is nowhere dense in Z . Proposition 4.1.
The composition of two weakly discontinuous maps is weakly discontinuous.Proof.
Let f : X → Y , g : Y → Z be two weakly discontinuous maps. To show that g ◦ f is weaklydiscontinuous, it suffices, given a non-empty subspace A ⊂ X to find a non-empty open subset U ⊂ A such that g ◦ f | U is continuous. The weak discontinuity of f yields a non-empty open set V ⊂ A suchthat f | V is continuous. The weak discontinuity of g yields a non-empty open subset W ⊂ f ( V ) such that g | W is continuous. By the continuity of f | V , the preimage U = ( f | V ) − ( W ) is open in V and hence in A . Finally, the continuity of the functions f | U and g | f ( U ) imply the continuity of g ◦ f | U . (cid:3) Proposition 4.2.
The composition g ◦ f : X → Z of a weakly discontinuous map f : X → Y and ascatteredly continuous map g : Y → Z is scatteredly continuous.Proof. Given a non-empty subspace A ⊂ X we should find a continuity point of g ◦ f | A . The weakdiscontinuity of f implies the existence of a non-empty open set V ⊂ A such that f | V is continuous.The scattered continuity of g implies that the existence of a continuity point y ∈ f ( V ) of the restriction g | f ( V ). Then any point x ∈ ( f | V ) − ( y ) is a continuity point of g ◦ f | A . (cid:3) Remark 4.3.
In light of Proposition 4.2 it is interesting to note that the composition g ◦ f : X → Z of a scatteredly continuous map f : X → Y and a weakly discontinuous map g : Y → Z need not bescatteredly continuous. A suitable pair of maps f, g is given in Example 2.4 (it is easy to check that thecharacteristic map χ Q : R Q → { , } is weakly discontinuous).In some other terms the following important result on interplay between scatteredly continuous andweakly discontinuous maps was established in [AB] and [BM]. We present a proof here for convenienceof the reader. Theorem 4.4.
A map f : X → Y from a topological space X into a (regular) topological space Y isscatteredly continuous if (and only if ) f is weakly discontinuous. Weakly discontinuous functions are refered to as Baire ∗ N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 5
Proof.
The “if” part is trivial. To prove the “only if” part, assume that f : X → Y is a scatteredlycontinuous map into a regular space Y . Since the scattered continuity is preserved by taking restrictions,it suffices to check that the set D ( f ) of discontinuity points of f is nowhere dense. This is equivalentto saying that the interior of the set C ( f ) of continuity points of f meets each non-empty open set U ⊂ X . Assuming the converse we would get U ⊂ D ( f ). Fix a continuity point x ∈ D ( f ) ∩ U ofthe restriction f | D ( f ) ∩ U (it exists because of the scattered continuity of f ). Since x fails to be acontinuity point of f , there is a neighborhood Of ( x ) ⊂ Y of the point f ( x ) such that f ( Ox ) Of ( x )for any neighborhood Ox of the point x in X . By the regularity of Y , find a neighborhood O ∗ f ( x )of f ( x ) such that O ∗ f ( x ) ⊂ Of ( x ). By the continuity of the restriction f | D ( f ) ∩ U at x there is aneighborhood Ox ⊂ U of x in X such that f ( Ox T D ( f )) ∈ O ∗ f ( x ). It follows from the choice ofthe neighborhoods Of ( x ) and O ∗ f ( x ) that f ( Ox ) O ∗ f ( x ) and hence there is a point x ∈ Ox with f ( x ) / ∈ O ∗ f ( x ). It follows from f ( D ( f ) ∩ Ox ) ⊂ O ∗ f ( x ) that x ∈ C ( f ). Consequently, there isa neighborhood Ox ⊂ Ox such that f ( Ox ) ⊂ Y \ O ∗ f ( x ). Since D ( f ) ⊃ U ⊃ Ox , there is a point x ∈ D ( f ) ∩ Ox . For this point we get f ( x ) ⊂ f ( Ox ∩ D ( f )) ∩ f ( Ox ) ⊂ O ∗ f ( x ) ∩ ( Y \ O ∗ f ( x )) = ∅ ,which is a contradiction. (cid:3) The following example shows that the regularity of Y is essential in the previous theorem. Example 4.5.
The identity map f : R → R Q from Example 2.4 shows that a scatteredly continuousmap to a non-regular space need not be weakly discontinuous.Unifying Proposition 4.2 with Theorem 4.4 we get a promised Proposition 4.6. If f : X → Y and g : Y → Z are scatteredly continuous maps and the space Y isregular, then the composition g ◦ f : X → Z is scatteredly continuous. Like the scattered continuity, the weak discontinuity also can be measured by an ordinal index. Namely,given a weakly discontinuous map f : X → Y consider the decreasing transfinite sequence ( ˜ D α ( f )) α ofclosed subsets of X defined recursively: ˜ D ( f ) = X and ˜ D α ( f ) = T β<α cl X (cid:0) D ( f | ˜ D β ( f )) (cid:1) . The transfiniteseries ( ˜ D α ( f )) α will be called the weak discontinuity series of f . The weak discontinuity of f impliesthat ˜ D α ( f ) = ∅ for some ordinal α . The smallest such an ordinal α is denoted by wd( f ) and is called the index of weak discontinuity of f . The following characterization of the weak discontinuity can be provedby analogy with Proposition 3.1. Proposition 4.7.
A map f : X → Y is weakly discontinuous if and only if there is a vanishing series ( X α ) α ≤ β of closed subsets of X such that X α +1 ⊃ D ( f | X α ) for every ordinal α < β . The smallest length β of such a series is equal to wd( f ) , the index of weak discontinuity of f . The index of weak discontinuity is related to the index of scattered continuity sc( f ) and to the hered-itary Lindel¨of number hl( X ) of X as follows. We recall that hl( X ) = sup { l ( Y ) : Y ⊂ X } where l ( Y ), the Lindel¨of number of a space Y , is the smallest cardinal κ such that each open cover of Y has a subcoverof size ≤ κ . Proposition 4.8. If f : X → Y is a weakly discontinuous map, then sc( f ) ≤ wd( f ) < hl( X ) + .Proof. The inequality wd( f ) < hl( X ) + follows from that fact that any strictly decreasing transfinitesequence of closed subsets of a topological space X has length < hl ( X ) + . The inequality sc( f ) ≤ wd( f )follows from the inclusions D α ( f ) ⊂ ˜ D α ( f ) that can be proved by induction on α . Alternatively it canbe derived from Proposition 3.1. (cid:3) Question 4.9.
Is sc( f ) = wd( f ) for any scatteredly continuous map f : [0 , → R ? Question 4.10.
Is sc( f ) < hl( X ) + for every scatteredly continuous map f : X → Y ?Propositions 4.1 and 4.2 can be quantified with help of ordinal indices sc( · ) and wd( · ) as follows. Firstwe need to recall the definition of ordinal multiplication, which can be introduced by recursion: α · α · ( β + 1) = α · β + α . Proposition 4.11. If f : X → Y is a weakly discontinuous map and g : Y → Z is a scatteredlycontinuous map, then sc( g ◦ f ) ≤ sc( g ) · wd( f ) . TARAS BANAKH AND BOGDAN BOKALO
Proof.
Observe that for every ordinal α < wd( f ) the function f is continuous at the set ˜ C α ( f ) =˜ D α ( f ) \ ˜ D α +1 ( f ), which is open and dense in ˜ D α ( f ).Given ordinals α < wd( f ) and β < sc( g ) consider the sets D α,β = ˜ C α ( f ) ∩ f − ( D β ( g )) and C α,β = ˜ C α ( f ) ∩ f − (cid:0) C ( g | D β ( g )) (cid:1) = D α,β \ D α,β +1 and note that the restriction g ◦ f | D α,β is continuous at each point of C α,β .On the product wd( f ) × sc( g ) = { ( α, β ) : α < wd( f ) , β < sc( g ) } consider the lexicographic ordering:( α, β ) ≤ ( α ′ , β ′ ) iff either α < α ′ or α = α ′ and β ≤ β ′ . Endowed with this ordering the productwd( f ) × sc( g ) is order isomorphic to the ordinal product sc( g ) · wd( f ). It is clear that the successor of apair ( α, β ) in (wd( f ) × sc( g ) , ≤ ) is the pair ( α, β + 1).For ordinals α < wd( f ) and β < sc( g ) let X ( α,β ) = ∪{ C α ′ ,β ′ : ( α, β ) ≤ ( α ′ , β ′ ) ∈ wd( f ) × sc( g ) } . Thus we obtain a vanishing series ( X ( α,β ) ) ( α,β ) of subsets of X . Since the lexicographic ordering ofwd( f ) × sc( g ) has order type of the ordinal product sc( g ) · wd( f ), the inequality sc( g ◦ f ) ≤ sc( g ) · wd( f )will follow from Proposition 3.1 as soon as we prove that for each ( α, β ) ∈ wd( f ) × sc( g ) the restriction g ◦ f | X ( α,β ) is continuous at each point x of X ( α,β ) \ X ( α,β +1) .Since x ∈ X ( α,β ) \ X ( α,β +1) = C α,β ⊂ X \ ˜ D α +1 ( f ), the continuity of g ◦ f | X ( α,β ) at x will follow assoon as we prove the continuity of g ◦ f restricted to the set Z α,β = X ( α,β ) \ ˜ D α +1 ( f ) at x . For thisobserve that Z α,β = S β ′ ≥ β C α,β ′ = ˜ C α ( f ) ∩ f − ( D β ( g )) and f ( x ) ∈ f ( C α,β ) ⊂ C ( g | D β ( g )). Now thecontinuity of f on the set Z α,β and the inclusion f ( Z α,β ) ⊂ D β ( g ) imply that g ◦ f | Z α,β is continuous as x . (cid:3) By analogy one can prove
Proposition 4.12. If f : X → Y and g : Y → Z are two weakly discontinuous maps then theircomposition g ◦ f is weakly discontinuous and wd( g ◦ f ) ≤ wd( g ) · wd( f ) . Corollary 4.13. If f : X → Y is scatteredly continuous (and weakly discontinuous) map and Z ⊂ X ,then sc( f | Z ) ≤ sc( f ) (and wd( f | Z ) ≤ wd( f ) ). Scatteredly continuous and piecewise continuous maps
In [Vino] V.A.Vinokurov proved that a map f : X → Y defined on a (complete) metrizable space X isweakly discontinuous (if and) only if f is piecewise continuous. We recall that a map f : X → Y betweentopological spaces is piecewise continuous if X has a countable closed cover C such that for every C ∈ C the restriction f | C is continuous. In [BKMM] the Vinokurov’s results was generalized to maps definedon perfectly paracompact (hereditarily Baire) spaces: Theorem 5.1.
A map f : X → Y from a perfectly paracompact (and hereditarily Baire) space X to atopological space Y is weakly discontinuous (if and) only if f is piecewise continuous. We recall that a topological space X is called • hereditarily Baire if each closed subspace F ⊂ X is Baire (in the sense that the intersection of asequence of open dense subsets of F is dense in F ); • perfectly paracompact if X is paracompact and each closed subset of X is of type G δ in X .The class of perfectly paracompact spaces is quite large: besides metrizable spaces this class includes allhereditarily Lindel¨of spaces and all stratifiable spaces, see [Bor].In this section we generalize Theorem 5.1 and prove its quantitative version.First note that the piecewise continuity of a map f : X → Y can be expressed in terms of the closeddecomposition number dec c ( f ) equal to the smallest size |C| of a closed cover C of X such that f | C iscontinuous for each C ∈ C , see [Sol]. Note that a map f is continuous iff dec c ( f ) = 1 and piecewisecontinuous iff dec c ( f ) ≤ ℵ .The following easy proposition (whose proof is left to the reader) shows that the continuity coveringnumber behaves nicely with respect to compositions. N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 7
Proposition 5.2.
For any maps f : X → Y and g : Y → Z we get dec c ( g ◦ f ) ≤ max { dec c ( g ) , dec c ( f ) } . It turns out that the closed decomposition number dec c ( f ) of a weakly discontinuous map f : X → Y can be estimated from above by the index of weak discontinuity wd( f ) of f and the large pseudocharacterΨ( X ) of X . By definition, for a topological T -space X the large pseudocharacter Ψ( X ) is equal to thesmallest cardinal κ such that each closed subset F ⊂ X can be written as the intersection ∩U of a family U consisting ≤ κ open subsets of X . It is easy to see that Ψ( X ) ≤ hl( X ) for every regular space X . Proposition 5.3. If f : X → Y is a weakly discontinuous map from a T -space X , then dec c ( f ) ≤ max {| wd( f ) | , Ψ( X ) } .Proof. Given a weakly discontinuous map f : X → Y consider the weak discontinuity series ( ˜ D α ( f )) α< wd( f ) of f . Let κ = max {| wd( f ) | , Ψ( X ) } . It follows from Ψ( X ) ≤ κ that for every α < wd( f ) the set˜ D α ( f ) \ ˜ D α +1 ( f ) can be written as the union ∪C α of a family of closed subsets of X with |C α | ≤ κ . Thecontinuity of f on ˜ D α ( f ) \ ˜ D α +1 ( f ) implies the continuity of f on each set C ∈ C α . Then C = S α< wd( f ) C α is a closed cover of X such that | C | ≤ κ and for every C ∈ C the restriction f | C is continuous. Thisproves the inequality dec c ( f ) ≤ κ = max {| wd( f ) | , Ψ( X ) } . (cid:3) In fact, the index of weak discontinuity of f in Proposition 5.3 can be replaced by the paracompactnessnumber par( X ) of X .We recall that a topological space X is paracompact if each open cover of X can be refined to a locallyfinite open cover. According to [En, 5.1.11], this is equivalent to saying that each open cover of X canbe refined to a locally finite closed cover. In light of this characterization it is natural to introduce acardinal invariant measuring the paracompactness degree of a topological space X .By the paracompactness number par( X ) of a topological space X we understand the smallest cardinal κ such that each open cover of X can be refined by an closed cover F of X that can be written as theunion F = S α<κ F α of κ many locally finite families F α of closed subsets of X . Hence a topologicalspace X is paracompact if and only if par( X ) ≤
1. It is easy to see that par( X ) ≤ l ( X ) for every regularspace X . Theorem 5.4. If f : X → Y is a weakly discontinuous map, then dec c ( f ) ≤ max { par( X ) , Ψ( X ) } .Proof. Given a weakly discontinuous map f : X → Y , consider the weak discontinuity series ( ˜ D α ( f )) α< wd( f ) of f .The proof of the inequality dec c ( f ) ≤ max { par( X ) , Ψ( X ) } will be done by induction on wd( f ). Ifwd( f ) ≤
1, then f is continuous and hence dec c ( f ) = 1 ≤ max { par( X ) , Ψ( X ) } . Assume that for someordinal α our theorem is proved for weakly discontinuous maps with index of weak discontinuity < α .Assume that f has wd( f ) = α and let λ = max { par( X ) , Ψ( X ) } .Consider the closed set C = T β<α ˜ D β ( f ) = T β +1 <α cl X (cid:0) D ( f | ˜ D β ( f )) (cid:1) and note that C = ˜ D α ( f ) if α is a limit ordinal and C = ˜ D γ ( f ) if α = γ + 1 is a successor ordinal. In any case the restriction f | C is continuous.Using the definition of the cardinal Ψ( X ) ≤ λ , find a family W = { W k : k < λ } of open sets of X with C = T k<λ W k .Take any ordinal k < λ and observe that par( X \ W k ) ≤ par( X ) ≤ λ . Then the open cover { X \ ˜ D β ( f ) : β < α } of X \ W k admits a closed refinement F k = S i<κ F k,i , where each family F k,i , i < λ , is locallyfinite in X \ W k .For every set F ∈ F k,i find an ordinal β < α with F ⊂ X \ ˜ D β ( f ). By induction on γ it can be shownthat ˜ D γ ( f | F ) ⊂ F ∩ ˜ D γ ( f ) for all ordinals γ . In particular, ˜ D β ( f | F ) = ∅ , which means that wd( f | F ) ≤ β < α . By the induction hypothesis, dec c ( f | F ) ≤ max { par( F ) , Ψ( F ) } ≤ max { par( X ) , Ψ( X ) } = λ . So,we can find a closed cover { C F,jk,i : j < λ } of F such that f | C F,jk,i is continuous for all α < κ . It followsfrom the local finity of F k,i that the family C jk,i = { C F,jk,i : F ∈ F k,i } is locally finite in X and hence theunion C jk,i = ∪C jk,i is a closed subset of X . Moreover, f | C jk,i is continuous. Then C = { C } ∪ S k,i,j<λ C jk,i is a closed cover of X of size ≤ λ witnessing that dec c ( f ) ≤ λ . (cid:3) Since max { par( X ) , Ψ( X ) } ≤ hl( X ) for every regular space X , Theorem 5.4 implies TARAS BANAKH AND BOGDAN BOKALO
Corollary 5.5. If f : X → Y is a weakly discontinuous map defined on a regular space X , then dec c ( f ) ≤ hl( X ) . Maps of stable first Baire class
Let us say that a sequence of maps ( f n : X → Y ) n ∈ ω stably converges to a map f : X → Y if for every x ∈ X there is n ∈ ω such that f m ( x ) = f ( x ) for all m ≥ n .A map f : X → Y between topological spaces is defined to be of the stable first Baire class if there isa sequence ( f n ) n ∈ ω of continuous functions from X to Y that stably converges to f .Maps of the stable first Baire class have been studied in [CL ], [CL ], [BKMM]. Proposition 6.1. If Y is a Hausdorff space, then each map f : X → Y of the stable first Baire class ispiecewise continuous.Proof. Choose a sequence ( f n : X → Y ) n ∈ ω of continuous functions that stably converges to f . For every n ∈ ω consider the closed subset X n = { x ∈ X : ∀ m ≥ n f m ( x ) = f n ( x ) } and note that f | X n ≡ f n | X n is continuous. The stable convergence of ( f n ) to f implies that X = S n ∈ ω X n which means that f is piecewise continuous. (cid:3) The converse statement to Proposition 6.1 is more subtle and holds under some assumptions on thespaces X and Y . For example, according to [BKMM] a piecewise continuous map f : X → Y is of thestable first Baire class if X is normal and Y is real line. The main result of this section asserts that thesame is true for any path-connected space Y ∈ σ AE( X ).To define the class σ AE( X ) we first recall the notion of an AE ( X )-space. Namely, we say that aspace Y is an absolute extensor for a space X and denote this by Y ∈ AE ( X ) if each continuous map f : A → Y defined on a closed subspace A ⊂ X has a continuous extension ˜ f : X → Y .The classical Urysohn Lemma says that R ∈ AE ( X ) for every normal space X . The Dugundji’sTheorem [Dug] says that each convex set Y in a locally convex space is an absolute extensor for anymetrizable space X (more generally, for any stratifiable space X , see [Bor]).Generalizing the notion of an AE ( X )-space to pairs ( Y, B ) of spaces B ⊂ Y we shall write ( Y, B ) ∈ AE ( X ) if each continuous map f : A → B defined on a closed subspace A ⊂ X has a continuous extension˜ f : X → Y . Hence Y ∈ AE ( X ) if and only if ( Y, Y ) ∈ AE ( X ).We define a space Y to belong to the class σ AE( X ) if Y has a countable cover { Y n : n ∈ ω } by closed G δ -sets such that ( Y, Y n ) ∈ AE ( X ) for all n ∈ ω . In particular, a space Y is a σ AE( X ) if it admits acountable cover { Y n : n ∈ ω } by closed G δ -subspaces Y n ∈ AE ( X ).Observe that the boundary Y = ∂I of the square I = [0 , is not an absolute extensor for X = I .On the other hand, for each proper subcontinuum B ⊂ ∂I the pair ( ∂I , B ) ∈ AE ( I ). Since ∂I canbe covered by two proper subcontinua, we get ∂I ∈ σ AE( I ). Example 6.2.
The Sierpi´nski carpet Y : N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 9 is an example of a Peano continuum that fails to be in σ AE( I ). The reason is that for each countableclosed cover C of Y the Baire Theorem yields a set C ∈ C with non-empty interior in Y . Then C containsa loop that is not contractible in Y , which means that ( Y, C ) / ∈ AE ( I ). Theorem 6.3.
Each piecewise continuous map f : X → Y from a normal space X to a path-connectedspace Y ∈ σ AE( X ) is of the stable first Baire class.Proof. Find a countable cover { Y n : n ∈ ω } of Y by non-empty closed G δ -subsets such that ( Y, Y n ) ∈ AE ( X ) for all n ∈ ω . In every set Y n choose a point y n ∈ Y n . Since Y is path-connected, for every n ∈ ω there is a continuous map γ n : [ n, n + 1] → Y such that γ ( n ) = y n and γ n ( n + 1) = y n +1 . The maps γ n compose a single continuous map γ : [0 , + ∞ ) → Y such that γ | [ n, n + 1] = γ n for all n ∈ ω . Let R + = [0 , + ∞ ) and denote by Gr ( γ ) = { ( γ ( t ) , t ) ∈ Y × R + : t ∈ R + } the graph of γ in Y × R + .Finally consider the subspace Z = Y × ω ∪ Gr ( γ ) of the product Y × R + and let p : Z → Y be therestriction of the projection Y × R + → Y . Define a map s : Y → Y × ω ⊂ Z letting s ( y ) = ( y, n ) if y ∈ Y n \ S i Example 6.4. Consider the characteristic map f : [0 , → { , } of the half-interval [0 , ). It is piecewisecontinuous but is not of the stable first Baire class (because continuous maps from [0 , 1] to { , } all areconstant). Yet, { , } ∈ σ AE( X ) for any space X .The condition Y ∈ σ AE( X ) also is essential in Theorem 7.1. Example 6.5. Let Y be the Sierpi´nski carpet in the square X = I , y ∈ Y be any point and f : X → Y be the piecewise continuous function defined by f ( x ) = x if x ∈ Y and f ( x ) = y if x / ∈ Y . We claimthat this function fails to be of the stable first Baire class. Assuming the converse, find a sequence( f n ) n ∈ ω of continuous functions that stably converges to f . For every n ∈ ω consider the closed set F k = { x ∈ X : ∀ m ≥ n f m ( x ) = f n ( x ) } and note that S k ∈ ω F k = X . The Baire Theorem implies thatfor some k ∈ ω the set F k ∩ Y has non-empty interior in Y . Then there is a map g : ∂I → F k ∩ Y thatcannot be extended to a continuous map ¯ g : I → Y . On the other hand, the map g has a continuousextension ˜ g : I → X . Then f n ◦ ˜ g : I → Y is a continuous extension of g = f ◦ g = f n ◦ ˜ g | ∂I whichcontradicts the choice of the map g . Scatteredly continuous and G δ -measurable maps In this section we reveal the relation of scatteredly continuous maps to G δ -measurable maps.We define a map f : X → Y to be G δ -measurable if for every open set U ⊂ Y the preimage f − ( U ) isof type G δ in X (this is equivalent to saying that for every closed subset F ⊂ Y the preimage f − ( F ) isof type F σ in X ).The following proposition is immediate. Proposition 7.1. A piecewise continuous map f : X → Y is G δ -measurable. According to [Vino] or [Kir], each G δ -measurable map f : X → Y from a complete metric space X toa regular space Y is weakly discontinuous. In Theorem 7.4 below we shall show that this is still true formaps from hereditarily Baire spaces X with the Preiss-Simon property.We define a topological space X to be Preiss-Simon at a point x ∈ X if for any subset A ⊂ X with x ∈ A there is a sequence ( U n ) n ∈ ω of non-empty open subsets of A that converges to x in the sense thateach neighborhood of x contains all but finitely many sets U n . By P S ( X ) we denote the set of points x ∈ X at which X is Preiss-Simon. A topological space X is called a Preiss-Simon space if P S ( X ) = X (that is X is Preiss-Simon at each point x ∈ X ).It is clear that each first countable space is Preiss-Simon and each Preiss-Simon space is Fr´echet-Urysohn. A less trivial fact due to D.Preiss and P.Simon [PS] asserts that each Eberlein compact spaceis Preiss-Simon.The proof of Theorem 7.4 is rather difficult and requires some preliminary work.A base B of the topology of a space X will be called regular if for each open set U ⊂ X and a point x ∈ U there are two sets V, W ∈ B with x ∈ V ⊂ X \ W ⊂ U . Such a regular base N will be called countably additive if the union ∪C of any countable subfamily C ⊂ B belongs to B .Observe that a topological space X admits a regular base if and only if X is regular. Note also thatthe family of all functionally open subsets of a Tychonov space X forms a regular countably additivebase of the topology of X . We recall that a subset U ⊂ X is called functionally open if U = f − ( V ) forsome continuous map f : X → R and some open set V ⊂ R .We define a map f : X → Y to be weakly G δ -measurable if there is a regular countably additive base B of the topology of Y such that for every U ∈ B the preimage f − ( U ) is of type G δ in X .Since the topology of each regular space forms a regular countably additive base, we conclude thateach G δ -measurable map f : X → Y into a regular topological space Y is weakly G δ -measurable.Let us say that a map f : X → Y between topological spaces is almost continuous ( quasi-continuous )at a point x ∈ X if for any neighborhood Oy ⊂ Y of the point y = f ( x ) the (interior of the) set f − ( Oy )is dense in some neighborhood of the point x in X . By AC ( f ) (resp. QC ( f ) ) we shall denote the set ofpoint of almost (resp. quasi-) continuity of f . Lemma 7.2. Let f : X → Y be a weakly G δ -measurable map from a Baire space X to a regular space Y . Then (1) AC ( f ) = QC ( f ) . (2) If D is dense in X and f | D has no continuity point, then D \ AC ( f ) also is dense in X . (3) For any countable dense set D ⊂ X there is a point y ∈ f ( D ) such that for every neighborhood Oy of y the preimage f − ( Oy ) has non-empty interior in X . (4) The family { Int f − ( y ) : y ∈ Y } is disjoint.Proof. Let B be a regular countably additive base of the topology of Y such that for every U ∈ B thepreimage f − ( U ) is of type G δ in X .1. The inclusion QC ( f ) ⊂ AC ( f ) is trivial. To prove that AC ( f ) ⊂ QC ( f ), take any point x ∈ AC ( f ).To show that x ∈ QC ( f ), take any neighborhood Oy ⊂ Y of the point y = f ( x ). We should check thatthe interior of f − ( Oy ) is dense in some neighborhood of x . By the regularity of the base B , there are sets V y, W y ∈ B such that y ∈ V y ⊂ Y \ W y ⊂ Oy . Then the preimages f − ( V y ) and f − ( W y ) are of type G δ and F σ in X , respectively. Since x ∈ AC ( f ), the closure of f − ( V y ) contains an open neighborhood Ox of x . We claim that the closure of the interior of f − ( Oy ) contains Ox . This will follow as soon as,given a non-empty open subset U ⊂ Ox we find a non-empty open subset U ′ ⊂ U ∩ f − ( Oy ). Observe N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 11 that the set U ∩ f − ( V y ), being a dense G δ -set in the Baire space U , is not meager in U . Consquently,the F σ -subset U ∩ f − ( W y ) ⊃ O x ∩ f − ( V y ) is non-meager in U too. Now the Baire Theorem yields anon-empty open subset U ′ ⊂ U ∩ f − ( W y ) ⊂ U ∩ f − ( Oy ) . 2. Assume that D ⊂ X is dense and f | D has no continuity point. Given a point x ∈ D , and aneighborhood O x of x we should find a point x ′ ∈ O x ∩ D \ AC ( f ). If x / ∈ AC ( f ), then we can take x ′ = x .So we assume that x ∈ AC ( f ) and hence x ∈ QC ( f ) by the preceding item. Since x is a discontinuitypoint of f | D , there is a neighborhood O f ( x ) of f ( x ) such that f ( D ∩ U x ) O f ( x ) for every neighborhood U x of x . Using the regularity of Y choose a neighborhood U f ( x ) ⊂ Y of f ( x ) with U f ( x ) ⊂ O f ( x ) . Since x is a quasi-continuity point of f , the closure of the interior of the preimage f − ( U f ( x ) ) contains someneighborhood W x of x . By the choice of O f ( x ) , we can find a point x ′ ∈ D ∩ O x ∩ W x with f ( x ′ ) / ∈ O f ( x ) .Consider the neighborhood O f ( x ′ ) = Y \ U f ( x ) of f ( x ′ ) and observe that W x ∩ f − ( O f ( x ′ ) ) is a nowheredense subset of Ox (because it misses the interior of f − ( U f ( x ) ) which is dense in W x ). This witnessesthat x ′ / ∈ AC ( f ).3. Given a countable dense subset D ⊂ X , we should find a point y ∈ f ( D ) such that for everyneighborhood Oy the preimage f − ( Oy ) has non-empty interior in X .Assume conversely that each point y ∈ f ( D ) has a neighborhood Oy such that the preimage f − ( Oy )has empty interior in X . Using the regularity of B , for every y ∈ f ( D ) find two sets V y, W y ∈ B with y ∈ V y ⊂ Y \ W y ⊂ Oy . It follows that f − ( V y ) is of type G δ while f − ( Y \ W y ) of type F σ in X .Being an F σ -set with empty interior in the Baire space X , the set f − ( Y \ W y ) is meager in X . Thenthe union U = [ y ∈ f ( D ) f − ( V y ) ⊃ D is a dense meager set in X too.On the other hand, the set U is a G δ -subset of X , being the preimage U = f − ( V ) of the subset V = S y ∈ f ( D ) V y that belongs to B because of the countable additivity of B . By the Baire Theorem, thedense G δ -subset U of the Baire space X cannot be meager, which is a contradiction.4. Assuming that the family { Int f − ( y ) : y ∈ Y } is not disjoint, find two distinct points y, z ∈ Y suchthat the intersection W = Int f − ( y ) ∩ Int f − ( z )is not empty. Observe that the sets W ∩ f − ( y ) and W ∩ f − ( z ) both are dense in W .By the Hausdorff property of Y the points y, z have disjoint open neighborhoods Oy, Oz ∈ B . Thechoice of B guarantees that the sets W ∩ f − ( Oy ) and W ∩ f − ( Oz ) are dense disjoint G δ -sets in theBaire space W , which is forbidden by the Baire Theorem. (cid:3) Lemma 7.3. Let f : X → Y be a weakly G δ -measurable map from a Baire space X to a regular space Y and D be a countable dense subset of X such that f | D has no continuity point. (1) For any finite subset F ⊂ Y there is a dense subset Q ⊂ D \ f − ( F ) in X such that f | Q has nocontinuity point. (2) For any sequence ( U n ) ∞ n =1 of non-empty open subsets of X there are an infinite subset I ⊂ N andsequences ( V n ) n ∈ I and ( W n ) n ∈ I of pairwise disjoint non-empty open sets in X and Y , respectively,such that V n ⊂ U n ∩ f − ( W n ) for all n ∈ I . (3) If D ⊂ P S ( X ) , then there is a countable first countable subspace Q ⊂ D having no isolated pointsand such that the restriction f | Q is a bijective map whose image f ( Q ) is a discrete subspace of Y .Proof. Fix a regular countably additive base B of the topology of Y such that for every U ∈ B thepreimage f − ( U ) is of type G δ in X .1. Take any finite subset F = { y , . . . , y n } in Y and for every i ≤ n let D i = D ∩ f − ( y i ) and W i = Int D i . First we show that the complement D \ f − ( F ) is dense in X . Assuming the converse, we can find anon-empty open set U ⊂ X such that U ∩ D ⊂ f − ( F ). We claim that U ∩ W i ∩ W j = ∅ for some distinctnumbers i, j ≤ n . Assuming conversely that the family { U ∩ W i } ni =1 is disjoint, we would conclude that W ∩ U ∩ D ⊂ D and hence the restriction f | D ∩ W ∩ U , being a constant map to y , is continuous.But this contradicts the fact that f | D has no continuity point. So, W = U ∩ W i ∩ W j is not emptyfor some distinct i, j ≤ n . Then D i ∩ W and D j ∩ W are disjoint dense subsets in W . Using theHausdorff property fo Y , take two disjoint neighborhoods Oy i , Oy j ∈ B . Then the dense disjoint sets G i = W ∩ f − ( Oy i ) ⊃ W ∩ D i and G j = W ∩ f − ( Oy j ) ⊃ W ∩ D j are of type G δ in the Baire space W , which contradicts the Baire Theorem. This completes the proof of the density of D \ f − ( F ).It follows that the set Q = (cid:0) D \ f − ( F ) (cid:1) \ n [ i =1 W i \ W i is dense in X . We claim that the restriction f | Q has no continuity point. Assume conversely that somepoint x ∈ Q is a continuity point of the restriction f | Q . If x / ∈ S ni =1 W i , then the discontinuity of themap f | D at x implies the discontinuity of f | Q at x . So, x ∈ W i for some i ≤ n . Let y = f ( x )and observe that y = y i (because x / ∈ f − ( F )). By the Hausdorff property of Y the points y and y i have disjoint open neighborhoods Oy , Oy i ∈ B . By the continuity of f | Q at x , there is an openneighborhood Ox ⊂ W i of x such that f ( Ox ∩ Q ) ⊂ Oy . It follows that the sets G = Ox ∩ f − ( Oy )and G i = Ox ∩ f − ( Oy i ) are of type G δ in Ox . The density of the set D \ f − ( F ) implies the densityof the set G ⊃ Ox ∩ Q = ( D \ f − ( F )) ∩ W i in Ox . On the other hand, the definition of the set W i implies that the intersection f − ( y i ) ∩ W i isdense in W i and hence G i ⊃ f − ( y i ) ∩ Ox is dense in Ox . Thus we have found two disjoint dense G δ -sets in the Baire space Ox which is forbiddenby the Baire Theorem.2. Let ( U n ) ∞ n =1 be a sequence of non-empty open subsets of X . Applying Lemma 7.2(3) to the map f | U and the dense subset D ∩ U , find a point y ∈ f ( D ∩ U ) such that for each neighborhood Oy the preimage U ∩ f − ( Oy ) has non-empty interior. By induction, for every n ∈ N we shall find a point y n ∈ f ( D ) \ { y i : i < n } such that for every neighborhood Oy n the set U n ∩ f − ( Oy n ) has non-emptyinterior.Assuming that for some n the points y , . . . , y n − have beeing chosen, we shall find a point y n . It followsthat the intersection D ∩ U n is a countable dense subset of U n such that f | D ∩ U n has no continuitypoint. Applying the preceding item, we can find a dense subset Q ⊂ D ∩ U n \ f − ( { y , . . . , y n − } ) in U n such that the restriction f | Q has no continuity point. Applying Lemma 7.2(3) to the map f | U n and thedense subset Q , find a point y n ∈ f ( Q ) ⊂ f ( D ) \ { y i : i < n } such that for each neighborhood Oy n thepreimage U n ∩ f − ( Oy n ) has non-empty interior. This completes the inductive construction.The space { y n : n ∈ N } , being infinite and regular, contains an infinite discrete subspace { y n : n ∈ I } .By induction, we can select pairwise disjoint open neighborhoods W n ⊂ Y , n ∈ I , of the points y n . Forevery n ∈ I the set U n ∩ f − ( W n ) contains a non-empty open set V n by the choice of the point y n . Thenthe set I ⊂ N and sequences ( V n ) n ∈ I , ( W n ) n ∈ I satisfy our requirements.3. Assume that D ⊂ P S ( X ). Applying Lemma 7.2(2), we get that D \ AC ( f ) is dense in X .By induction on the tree ω <ω we shall construct sequences ( x s ) s ∈ ω <ω of points of the set D \ AC ( f ),and sequences ( V s ) s ∈ ω <ω and ( U s ) s ∈ ω <ω , ( W s ) s ∈ ω <ω of sets so that the following conditions hold for everyfinite number sequence s ∈ ω <ω :(a) V s is an open neighborhood of the point x s in X ;(b) W s ⊂ U s are open neighborhoods of f ( x s ) in Y ;(c) f ( V s ) ⊂ U s ;(d) V s ˆ n ⊂ V s and U s ˆ n ⊂ U s for all n ∈ ω ;(e) the sequence ( V s ˆ n ) n ∈ N converges to x s ;(f) W s ∩ U s ˆ n = ∅ = U s ˆ n ∩ U s ˆ m for all n = m in ω . N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 13 We start the induction letting V ∅ = X , U ∅ = Y and x ∅ be any point of D \ AC ( f ).Assume that for a finite sequence s ∈ ω <ω the point x s ∈ D \ AC ( f ) and open sets V s ⊂ X and U s ⊂ Y with x s ∈ V s and f ( V s ) ⊂ U s have been constructed. Since f | V s fails to be almost continuousat x s , there is a neighborhood W s ⊂ U s of f ( x s ) such that the closure of the preimage f − ( W s ) is nota neighborhood of x s in X . This fact and the Preiss-Simon property of X at x s allows us to constructa sequence ( V ′ k ) k ∈ ω of open subsets of V s \ cl X (cid:0) f − ( W s ) (cid:1) that converges to x s in the sense that eachneighborhood of x contains all but finitely many sets V ′ k . Applying item 2 to the map f | V s : V s → U s ,we can find an infinite subset N ⊂ ω and a sequence ( U ′ k ) k ∈ N of pairwise disjoint open sets of U s suchthat each set f − ( U ′ k ) ∩ V ′ k , k ∈ N , has non-empty interior in X . Let N = { k n : n ∈ ω } be the increasingenumeration of the set N .For every n ∈ ω let U s ˆ n = U ′ k n \ W s , V s ˆ n be a non-empty open subset in f − ( U k n ) ∩ V ′ k n and x s ˆ n ∈ V s ˆ n ∩ D \ AC ( f ) be any point (such a point exists because of the density of D \ AC ( f ) in X ).One can check that the points x s ˆ n , n ∈ ω and sets W s , V s ˆ n , U s ˆ n , n ∈ ω satisfy the requirements of theinductive construction.After completing the inductive construction, consider the set Q = { x s : s ∈ ω <ω } and note that it isfirst countable and has no isolated point, f | Q is bijective and f ( Q ) is a discrete subspace of Y . (cid:3) Now we are able to prove the promised Theorem 7.4. Each weakly G δ -measurable map f : X → Y from a hereditarily Baire Preiss-Simon space X to a regular space Y is scatteredly continuous.Proof. Fix a countably additive regular base B of the topology of Y such that f − ( U ) is of type G δ in X for every U ∈ B . Assuming that f is not scatteredly continuous we could find a subspace D ⊂ X such that the restriction f | D has no continuity point. Being Preiss-Simon, the space X has countabletightness. In this case we can additionally assume that the set D is countable, see Proposition 2.3.Applying Lemma 7.3(3) to the restriction f | D of f onto the Baire space D , we can find a countablesubset Q ⊂ D without isolated point such that f | Q is bijective and f ( Q ) is a discrete subspace of Y . Itis clear that f | Q has no continuity point. Observe that Z = Q is a Baire space, being a closed subspaceof the hereditarily Baire space X . Let g = f | Z and observe that for every U ∈ B the preimage g − ( U ) isof type G δ in Z .Since the set g ( Q ) = f ( Q ) is countable and discrete, for every y ∈ g ( Q ) we can select a neighborhood Oy ∈ B so small that the family { Oy : y ∈ g ( Q ) } is disjoint. The countable additivity of the base B implies that S y ∈ g ( Q ) Oy ∈ B and hence the preimage G = g − ( S y ∈ g ( Q ) Oy ) is a Baire space, being adense G δ -subset of the Baire space Z . Since G = S y ∈ g ( Q ) g − ( Oy ), the Baire Theorem guarantees thatfor some z ∈ g ( Q ) the preimage g − ( Oz ) is not meager in G . On the other hand, S y ∈ g ( Q ) \{ z } Oy ∈ B and hence the complement G \ g − ( Oz ) = g − ( S y ∈ g ( Q ) \{ z } Oy ) is a G δ -set in G . The set g − ( Oz ), beinga non-meager F σ -subset of the Baire space G , contains a non-empty open subset W of G . For this setwe get g ( W ∩ Q ) ⊂ Oz ∩ g ( Q ) = { z } , which means that g | Q is continuous at points of W ∩ Q . But this contradicts the fact that g | Q ≡ f | Q has no continuity point. (cid:3) Characterizing the scattered continuity Unifying Theorems 4.4, 5.1, 6.3, 7.4 and Propositions 6.1, 7.1, we obtain the following characterizationof scatteredly continuous maps: Theorem 8.1. For a map f : X → Y from a perfectly paracompact hereditarily Baire space X to aregular space Y the following conditions are equivalent: (1) f is scatteredly continuous; (2) f is weakly discontinuous; (3) f is piecewise continuous.If X is a Preiss-Simon space, then (1)–(3) are equivalent to (4) f is G δ -measurable; (5) f is weakly G δ -measurable;If Y is path-connected and belongs to σ AE( X ) , then (1)–(3) are equivalent to (6) f is of stable first Baire class. Theorem 8.1 has an interesting corollary describing the interplay between F σ - and G δ -measurablemaps. First observe that, for a space Y whose each open subset is of type F σ , each G δ -measurable map f : X → Y is F σ -measurable. The same effect can be achieved imposing some restrictions on the domain X of f . Corollary 8.2. Each (weakly) G δ -measurable map f : X → Y from a perfectly paracompact hereditarilyBaire Preiss-Simon space X to a regular space Y is F σ -measurable. Question 8.3. Let f : X → Y be a G δ -measurable function from a perfectly paracompact hereditarilyBaire space X to a regular space Y . Is f F σ -measurable?9. Some Examples The requirement that X is hereditarily Baire is necessary in Theorem 8.1. Proposition 9.1. If a perfectly paracompact space X is not hereditarily Baire, then there is a bijectivepiecewise continuous map f : X → Y that is not scatteredly continuous.Proof. Since the space X is not hereditarily Baire, there is a closed subspace F ⊂ X of the first Bairecategory in itself. Write F = S n ∈ ω F n as the union of an increasing sequence ( F n ) n ∈ ω of closed nowheredense subsets of F with F = ∅ . Let Y be the topological sum of the family F = { X \ F, F n +1 \ F n : n ∈ ω } and f : X → Y be the identity map. The perfect paracompactness of X yields that each subset A ∈ F is of type F σ in X , which implies that f is piecewise continuous. On the other hand, the restriction f | F has no continuity point, which means that f fails to be scatteredly continuous. (cid:3) The above proposition shows that the equivalences (1) ⇔ (2) ⇔ (3) of Theorem 8.1 do not holdwithout the assumption that X is hereditarily Baire. On the other hand, the equivalence (3) ⇔ (4) doesnot require that assumption and hold, for example, for any map f : X → Y from an absolute F -Souslinspace X to a metrizable space Y , see [JR].Now we see that a G δ -measurable function f : X → Y between metrizable spaces is piecewise con-tinuous provided X is either hereditarily Baire (by Theorem 7.4) or absolute F -Souslin (by [JR]). Inthis situation it is natural to ask if any G δ -measurable function f : X → Y between metrizable spacesis piecewise continuous, see V.Vinokurov [Vino]. Below we give two (consistent) counterexamples to thisquestion.The first one uses Q -spaces. Those are topological spaces X such that each subset A ⊂ X is of type F σ in X . The Martin Axiom implies that each second countable space X of cardinality | X | < c is a Q -space, see [Miller, 4.2]. On the other hand, in some models of ZFC, each second countable Q -space isat most countable, see [Miller, 4.3]. Example 9.2. If X is an uncountable second countable Q -space, then any bijective map f : X → D toa discrete space D is G δ -measurable but not piecewise continuous. Moreover, dec( f ) = | X | .Here for a function f : X → Y by dec( f ) we denote the smallest cardinality |C| of a cover C of X such that f | C is continuous for every C ∈ C , see [Sol]. It is clear that dec( f ) ≤ dec c ( f ). The latterinequality can be strict: for the characteristic function χ Q : R → { , } of rationals we get dec( f ) = ℵ while dec c ( f ) = d , see [Sol].A typical example of an F σ -measurable function with dec( f ) > ℵ is the Pawlikowski function P :( ω + 1) ω → ω ω , which is the countable power of the bijection ω + 1 → ω sending every n ∈ ω to n + 1and ω to 0, see [CMPS], [Sol]. Here the ordinal ω + 1 = ω ∪ { ω } , endowed with the order topology, ishomeomorphic to a convergent sequence.Note that the range space D in Example 9.2 is not second countable. Our second counterexample tothe Vinokurov question has second countable range. It relies on the notion of a Lusin space. N SCATTEREDLY CONTINUOUS MAPS BETWEEN TOPOLOGICAL SPACES 15 Given a cardinal κ we define a topological space X to be κ -Lusin if each nowhere dense subset A of X has size | A | < κ . Uncountable ℵ -Lusin spaces are referred to as Lusin spaces, see [Miller, § b -Lusin spaces, where b is the smallest cardinality | A | of a subset A ⊂ N ω that does notlie in a σ -compact subset of N ω , see [Bl, § Example 9.3. Let P : ( ω + 1) ω → ω ω be the Pawlikowski function. For every b -Lusin subset L ⊂ ω ω of size | L | ≥ b and its preimage X = P − ( L ) the function f = P | X : X → ω ω is G δ -measurable but hasdec( f ) = | L | ≥ b . Proof. The G δ -measurability of f = P | X will follow as soon as we check that for every closed subset F ⊂ ω ω the set f − ( F ) = P − ( F ) ∩ X is of type F σ in X . First we consider the case of a nowhere dense F . In this case | F ∩ L | < b and hence the set A = f − ( F ) = P − ( F ) ∩ X has size | A | = | L ∩ F | < b .The space P − ( F ), being a Borel subset of ( ω + 1) ω is analytic and thus is the image of ω ω under asuitable continuous map g : ω ω → P − ( F ). Find a subset A ′ ⊂ ω ω of size | A ′ | = | A | with g ( A ′ ) = A .The definition of the cardinal b implies the existence of a σ -compact subset K ⊂ ω ω with K ⊃ A ′ . Then g ( K ) is a σ -compact subset of P − ( F ) containing the set A . Consequently, A = P − ( F ) ∩ X = g ( K ) ∩ X is of type F σ in X .If F is an arbitrary closed subset of ω ω , then take the interior U of F in ω ω and observe that P − ( U )is of type F σ in ( ω + 1) ω by the F σ -measurability of P . Then f − ( U ) = P − ( U ) ∩ X is of type F σ in X . Since F \ U is closed and nowhere dense in ω ω , the preimage f − ( F \ U ) is of type F σ in X too.Combining these two facts, we conclude that the preimage f − ( F ) = f − ( U ) ∪ f − ( F \ U ) is of type F σ in X , witnessing that f : X → ω ω is G δ -measurable.Next, we check that dec( f ) ≥ | L | . Let C be a cover of X such that |C| = dec( f ) and f | C is continuousfor every C ∈ C . Using the definition of the Pawlikowski function, it is easy to check that the image f ( C ) of each C ∈ C is nowhere dense in ω ω and hence | f ( C ) ∩ L | < b . It follows from | L | ≥ b and theregularity of the cardinal b [Bl, 2.4] that |C| ≥ | L | . (cid:3) In light of Example 9.3 it is important to have some conditions guaranteeing the existence of b -Lusin sets of cardinality ≥ b . Such conditions can be written in terms of the following three cardinalcharacteristics of the σ -ideal M of meager subsets of the Baire space ω ω :add( M ) = min {|A| : A ⊂ M , ∪A / ∈ M} , cov( M ) = min {|A| : A ⊂ M , ∪A = ω ω } , cof( M ) = min {|A| : A ⊂ M ∀ M ∈ M ∃ A ∈ A with M ⊂ A } The following proposition is a modification of Corollary 8.2.5 and Lemma 8.2.4 from [BJ]. Proposition 9.4. (1) If add( M ) = cof( M ) , then there is a b -Lusin subset L ⊂ ω ω of size | L | = b ; (2) If there is a b -Lusin subset L ⊂ ω ω of size | L | ≥ b , then cov( M ) ≥ b .Proof. 1. The Cicho´n diagram [Bl, § 5] guarantees add( M ) ≤ b ≤ cof( M ). Consequently, the equalityadd( M ) = cof( M ) implies add( M ) = b = cof( M ). Let { M α } α< b ⊂ M be a cofinal family of meagersubsets of ω ω . By the transfinite induction, for every α < b choose a point x α / ∈ S β ≤ α M β in ω ω . Sucha choice is always possible, since the latter union is meager (by add( M ) = b ). The set L = { x α : α < b } is the required b -Lusin subset of ω ω with | L | = b .2. Now assume that L ⊂ ω ω is a b -Lusin subset with | L | ≥ b . Fix a cover C ⊂ M of ω ω of size | C | = cov( M ). It follows that | L ∩ C | < b for all C ∈ C . Now the regularity of the cardinal b [Bl, 2.4]implies that cov( M ) = |C| ≥ b . (cid:3) By [Bl, 7.4], the Martin Axiom implies add( M ) = cof( M ) = c . On the other hand, the strict inequalitycov( M ) < b holds is the Laver’s model of ZFC, see [Bl, 11.7]. In this model no b -Lusin subset L ⊂ ω ω with | L | ≥ b exists. This shows that Example 9.3 has consistent nature. Question 9.5. Is it consistent with ZFC that each G δ -measurable function f : X → R defined on asubset X ⊂ R has dec( f ) ≤ ℵ ? dec c ( f ) ≤ ℵ ? Acknowledgement The authors express their sincere thanks to Lubomyr Zdomskyy for fruitful discussions resulting infinding Example 9.3. References [Ar] A.V. Arkhangelskii, The structure and classification of topological spaces and cardinal invariants , Uspekhi Mat.Nauk. :6 (1978), 29-84 (in Russian).[AB] A.V. Arkhangelskii, B.M. Bokalo, Tangency of topologies and tangential properties of topological spaces ,Tr.Mosk.Mat. Ob-va (1992), 160–185, (in Russian); English transl.: Trans. Mosk. Math. Soc. (1993),139-163.[B] R. Baire, Sur les fonctions de variables reelles , Annali di Mat. (1899), 1–123.[BKMM] T. Banakh, S. Kutsak, V. Maslyuchenko, O. Maslyuchenko, Direct and inverse problems of the Baire classifica-tions of integrals dependent on a parameter , Ukr. Mat. 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