A generalization of a Baire theorem concerning barely continuous functions
aa r X i v : . [ m a t h . GN ] J a n A GENERALIZATION OF A BAIRE THEOREM CONCERNING BARELYCONTINUOUS FUNCTIONS
OLENA KARLOVA , Abstract.
We prove that if X is a paracompact space, Y is a metric space and f : X → Y is a functionally fragmentedmap, then (i) f is σ -discrete and functionally F σ -measurable; (ii) f is a Baire-one function, if Y is weak adhesive andweak locally adhesive for X ; (iii) f is countably functionally fragmented, if X is Lindel¨off.This result generalizes one theorem of Rene Baire on classification of barely continuous functions. Introduction
A map f : X → Y between topological spaces X and Y is said to be- Baire-one , if it is a pointwise limit of a sequence of continuous maps f n : X → Y ;- (functionally) F σ -measurable or of the first (functional) Borel class , if the preimage f − ( V )of any open set V ⊆ Y is a union of a sequence of (functionally) closed sets in X ;- barely continuous , if the restriction f | F of f to any non-empty closet set F ⊆ X has a pointof continuity.Let us observe that the term ”barely continuous” belongs to Stephens [16]. However, barely contin-uous functions are also mentioned in literature as functions with the ”point of continuity property”(see, for instance, [13, 15]).Among many other characterizations of Baire-one functions, the following classical Baire’s the-orem is well-known [2]. Theorem A.
For a complete metric space X and a function f : X → R the following conditionsare equivalent:(1) f is Baire-one;(2) f is F σ -measurable;(3) f is barely continuous. Recall that a map f : X → Y between topological space X and a metric space Y is said to be fragmented , if for all ε > F ⊆ X there exists a relatively open set U ⊆ F such that diam f ( U ) < ε . The above notion was supposed by Jayne and Rogers [6] in connectionwith Borel selectors of certain set-valued maps.Evidently, every barely continuous map between a topological and a metric spaces is fragmented.Moreover, if X is a hereditarily Baire space, then every fragmented function is barely continuous.The property of baireness of X is essential: let us consider a function f : Q → R , f ( r n ) = 1 /n , where Q = { r n : n ∈ N } is the set of all rational numbers. Notice that f is fragmented and everywherediscontinuous.The next generalization of Baire’s theorem follows from [5, Corollary 7] and [1, Theorem 2.1]. Theorem B.
Let X be a hereditarily Baire paracompact perfect space, Y is a metric space and f : X → Y . The following conditions are equivalent: Mathematics Subject Classification.
Primary 54C30, 26A21; Secondary 54C50.
Key words and phrases. fragmented function, Baire-one function, F σ -measurable function, σ -dicrete function. , (i) f is F σ -measurable and σ -discrete;(ii) f is fragmented.Moreover, if Y is a convex subset of a Banach space, they are equivalent to:(iii) f is Baire-one. Let us observe that a similar result for Y = R was obtained by Mykhaylyuk [14].The next theorem was recently proved in [10, Theorem 10]. Theorem C. If X is a paracompact perfect space, Y is a metric contractible locally path-connectedspace and f : X → Y is fragmented, then f ∈ B ( X, Y ) . The aim of this note is to extend the above mentioned results on maps defined on paracompactspaces which are not necessarily perfect (recall that a topological space is perfect if every its opensubset is F σ ).The convenient tool of investigation of fragmented functions on non-perfect spaces is a conceptof functional fragmentability introduced in [11]. We prove a technical auxiliary result (Lemma 2)which connects regular families of functionally open sets in paracompact spaces with the notionof σ -discrete decomposability. As an application of this result we obtain (Theorem 3) that for aparacompact space X , a metric space Y and a functionally fragmented map f : X → Y the followingpropositions hold: (i) f is σ -discrete and functionally F σ -measurable; (ii) f is a Baire-one function,if Y is weak adhesive and weak locally adhesive for X ; (iii) f is countably functionally fragmented,if X is Lindel¨off. 2. Preliminaries
Let U = ( U ξ : ξ ∈ [0 , α ]) be a transfinite sequence of subsets of a topological space X . Then U is regular in X , if(a) each U ξ is open in X ;(b) U = ∅ , U α = X and U ξ ⊆ U η for all 0 ≤ ξ ≤ η < α ;(c) U γ = S ξ<γ U ξ for every limit ordinal γ ∈ [0 , α ]. Proposition 1. [12, Proposition 1]
Let X be a topological space, ( Y, d ) be a metric space and ε > .For a map f : X → Y the following conditions are equivalent:(1) f is ε -fragmented;(2) there exists a regular sequence U = ( U ξ : ξ ∈ [0 , α ]) in X such that diam f ( U ξ +1 \ U ξ ) < ε for all ξ ∈ [0 , α ) . If a sequence U satisfies condition (2) of the previous proposition, then it is called ε -associatedwith f and is denoted by U ε ( f ).We say that an ε -fragmented map f : X → Y is functionally ε -fragmented if U ε ( f ) can bechosen such that every set U ξ is functionally open in X . Further, f is functionally fragmented , if itis functionally ε -fragmented for each ε > f is (functionally) countably fragmented , if f is (functionally) fragmented and everysequence U ε can be chosen to be countable. GENERALIZATION OF A BAIRE THEOREM 3 A Lemma
Let A be a family of subsets of a topological space X . Then A is called • discrete , if each point x ∈ X has a neighborhood which intersects at most one set from A ; • strongly functionally discrete or, briefly, sfd-family , if there exists a discrete family ( U A : A ∈ A ) of functionally open subsets of X such that A ⊆ U A for every A ∈ A .Let us observe that every discrete family is strongly functionally discrete in collectionwise normalspace. Lemma 2. (cf. [3, Theorem 2])
Let U = ( U ξ : ξ ∈ [0 , α ]) be a regular family of functionally open setsin a paracompact space X . Then there exists a sequence ( F n ) n ∈ ω of families F n = ( F ξ,n : ξ ∈ [0 , α ]) such that(1) U ξ \ S η<ξ U η = S n ∈ ω F ξ,n for all ξ ∈ [0 , α ) ,(2) F n is an sfd-family in X for all n ∈ ω ,(3) F ξ,n is closed in X for all n ∈ ω and ξ ∈ [0 , α ) .Proof. For every ξ ∈ [1 , α ] we denote P ξ = U ξ \ S η<ξ U η . Since every P ξ is functionally G δ in X asa difference of two functionally open sets, we can choose a sequence ( G ξ,n ) n ∈ ω of functionally opensets such that P ξ = \ n ∈ ω G ξ,n for all ξ ∈ [1 , α ) and G ξ,n ⊆ U ξ for all ξ ∈ [1 , α ) , n ∈ ω. We put I = [ k ∈ ω ω k and define by the induction on k sequences ( U i : i ∈ I ) and ( V i : i ∈ I ) of open coverings of X suchthat(a) U ∅ = U ;(b) V i is a locally finite barycentric refinement of U i for all i ∈ ω k ;(c) for all i ∈ ω k and n ∈ ω we have U ( i,n ) = ( U ξ, ( i,n ) : ξ ∈ [0 , α )), where C ξ,i = { x ∈ X : St( x, V i ) ⊆ [ η<ξ U η } and U ξ, ( i,n ) = G ξ,n \ C ξ,i for all k ∈ ω . Let us observe that the existence of families V i follows from the paracompactness of X (see [4, Theorem 5.1.12]).Notice that C ξ,i ⊆ [ η<ξ U η , because V i is an open covering of X . Therefore, since ( P ξ : ξ ∈ [0 , α ]) is a partition of X , U ( i,n ) defined in (c) covers X for all n ∈ ω .For every x ∈ X we put µ ( x ) = min { ξ ∈ [0 , α ) : x ∈ U ξ } OLENA KARLOVA , and show that ∀ x ∈ X ∃ i ∈ I : St( x, V i ) ⊆ U µ ( x ) . (3.1)Assume to the contrary that there exists x ∈ X such that (3.1) is not true. Since each family V i is locally finite refinement of U , for every i ∈ I there is ξ i such that St( x, V i ) ⊆ U ξ i . Let ξ ( x ) = min { ξ i : i ∈ I } . Then ξ ( x ) > µ ( x ). Therefore, x P ξ ( x ) and we can take j ∈ ω such that x G ξ ( x ) ,j .From the definition of the sequence U ( i,j ) it follows that x U ξ ( x ) , ( i,j ) . Since St( x, V i ) ⊆ U ξ ( x ) ,we have x U ξ, ( i,j ) for all ξ > ξ ( x ). Therefore, x [ ξ ≥ ξ ( x ) U ξ, ( i,j ) (3.2)By (b) there exists β ∈ [0 , α ) such that St( x, V ( i,j ) ) ⊆ U β, ( i,j ) . It follows from (3.2) that β < ξ ( x ).The inclusion U β, ( i,j ) ⊆ U β contradicts to the choice of ξ ( x ).Let ( V i : i ∈ I ) = ( H n : n ∈ ω ). Now for all ξ ∈ [0 , α ) and n ∈ ω we put D ξ,n = { x ∈ P ξ : St( x, H n ) ⊆ U ξ } and F ξ,n = D ξ,n . We will show that P ξ = [ n ∈ ω F ξ,n for all ξ ∈ [0 , α ). Property (3.1) implies that P ξ ⊆ S n ∈ ω F ξ,n . Now assume that x ∈ F ξ,n for some ξ and n . Put O = St( x, H n ) ∩ U µ ( x ) . Then O ∩ D ξ,n = ∅ . Take any y ∈ O . Since y ∈ U µ ( x ) and y ∈ P ξ , µ ( x ) ≥ ξ . The inclusions St( y, H n ) ⊆ U ξ and y ∈ St( x, H n ) imply that x ∈ U ξ . Hence, µ ( x ) ≤ ξ .Therefore, µ ( x ) = ξ . Then x ∈ P ξ . Moreover, it follows that the family F n = ( F ξ,n : ξ ∈ [0 , α )) isdiscrete in X .Since X is paracompact, X is collectionwise normal, which implies that F n is strongly function-ally discrete family for all n ∈ ω . (cid:3) An application of Lemma to classification of fragmented functions
Let X be a topological space. Recall that a topological space Y is • an adhesive for X , if for any disjoint functionally closed sets A and B in X and for anytwo continuous maps f, g : X → Y there exists a continuous map h : X → Y such that h | A = f | A and h | B = g | B ; • a weak adhesive for X , if for any two points y, z ∈ Y and disjoint functionally closed sets A and B in X there exists a continuous map h : X → Y such that h | A = y i h | B = z ; • a locally weak adhesive for X , if for every y ∈ Y and every neighborhood V ⊆ Y of y thereexists a neighborhood U of y such that U ⊆ V and for every z ∈ U there exists a continuousmap h : X → V with h | A = y and h | B = z .It was proved in [9, Theorem 2.7] that any topological space Y is an adhesive for every stronglyzero dimensional space X ; a path-connected space Y is an adhesive for any compact space X eachpoint of which has a base of neighborhoods with discrete boundaries; Y is an adhesive for any space X if and only if Y is contractible. Moreover, it is easy to see that every (locally) path-connectedspace is a (locally) weak adhesive for any X .A family B of subsets of a topological space X is said to be a base for a map f : X → Y , if forevery open set V ⊆ Y there exists a subfamily B V of B such that f − ( V ) = S B ∈ B V B .A map f : X → Y is σ -discrete , if there is a sequence ( B n ) n ∈ ω of discrete families of sets in X such that the family S n ∈ ω B n is a base for f . GENERALIZATION OF A BAIRE THEOREM 5
Theorem 3.
Let X be a paracompact space, Y be a metric space and f : X → Y be a functionallyfragmented map. Then(1) f is σ -discrete and functionally F σ -measurable;(2) f is a Baire-one function, if Y is weak adhesive and weak locally adhesive for X ;(3) f is countably functionally fragmented, if X is Lindel¨off.Proof. For every n ∈ N we choose a family U /n ( f ) = ( U ξ,n : ξ ∈ [0 , α n ]) consisting of functionallyopen sets U ξ,n . We claim that the family P = S n ∈ N P n is a base for f , where P n = ( U ξ,n \ S η<ξ U η,n : ξ ∈ [0 , α n ]), n ∈ N . Indeed, fix an open set V in Y and take any x ∈ f − ( V ). Find n ∈ N such that an open ball B with the center at f ( x ) and radius 1 /n contains in V . Since P n is a partition of X , there exists ξ ∈ [0 , α n ] such that x ∈ P ξ,n = U ξ,n \ S η<ξ U η,n . Evidently, f ( P ξ,n ) ⊆ B ⊆ V .By Lemma 2 for every n ∈ N there exists a sequence ( F n,k ) k ∈ ω of families F n,k = ( F ξ,n,k : ξ ∈ [0 , α n ]) which satisfies conditions (1)–(3) of Lemma 2. Properties (1) and (2) imply that the family B = S k,n F n,k is a σ -discrete base for f consisting of closed sets. It follows that f is F σ -measurableand a σ -discrete map. Finally, [7, Proposition 2.6 (iv)] implies that f is functionally F σ -measurable.Property follows from 1) and [8, Theorem 3.2]. It is enough to show that every regular sequence consisting of functionally open sets in aLindel¨off space X is countable.Let U = ( U ξ : ξ ∈ [0 , α ]) be a regular covering of X by functionally open sets U ξ . There existsa sequence ( F n ) n ∈ ω of families in X such that conditions (1)–(3) of Lemma 2 are valid. Noticethat every family F n is at most countable, since it is discrete and X is Lindel¨off. We consider anenumeration { F k : k ∈ ω } of the family S n ∈ ω F n . Let ϕ : [0 , α ) → ω be a map, ϕ ( ξ ) = { k ∈ ω : F k ⊆ P ξ } . Since ( ϕ ( ξ ) : ξ ∈ [0 , ω )) is a family of mutually disjoint subsets of ω , it is at most countable. (cid:3) We do not know the answer to the following question.
Question 1.
Is it true that every fragmented Baire-one real-valued function defined on a paracom-pact Hausdorff space is functionally fragmented?
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