A generalization of a Baire theorem concerning barely continuous functions

TL;DR

This paper generalizes Baire's theorem by demonstrating that functionally fragmented maps from paracompact spaces to metric spaces possess specific measurability and fragmentation properties, extending the classification of barely continuous functions.

Contribution

It introduces a broader class of functions and establishes their measurability and fragmentation properties under general topological conditions, extending Baire's original theorem.

Findings

Functionally fragmented maps are $\sigma$-discrete and $F_\sigma$-measurable.

Such maps are Baire-one if the target space is weak adhesive.

They are countably functionally fragmented if the domain is Lindelöf.

Abstract

We prove that if $X$ is a paracompact space, $Y$ is a metric space and $f:X\to Y$ is a functionally fragmented map, then (i) $f$ is $\sigma$-discrete and functionally $F_\sigma$-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\"{o}ff. This result generalizes one theorem of Rene Baire on classification of barely continuous functions.

Equations

$$P_{\xi }=\bigcap _{n\in \omega }{G}_{\xi ,n}\text{mbox}forall\xi \in [1,\alpha )\text{mbox}and$$

$$G_{\xi ,n}\subseteq U_{\xi }\text{mbox}forall\xi \in [1,\alpha ),n\in \omega .$$

$$I=\bigcup _{k\in \omega }{\omega }^k$$

$$C_{\xi ,i}=\overline{\{x\in X:\mathrm{St}(x,{\mathcal{V}}_i)\subseteq \bigcup _{\eta <\xi }{U}_{\eta }\}}\text{mbox}and{U}_{\xi ,(i,n)}=G_{\xi ,n}\setminus C_{\xi ,i}$$

$$C_{\xi ,i}\subseteq \bigcup _{\eta <\xi }{U}_{\eta },$$

$$\mu (x)=\min \{\xi \in [0,\alpha ):x\in U_{\xi }\}$$

$$\forall x\in X\exists i\in I:\mathrm{St}(x,{\mathcal{V}}_i)\subseteq U_{\mu (x)}.$$

$$x\notin \bigcup _{\xi \ge \xi (x)}{U}_{\xi ,(i,j)}$$

$$D_{\xi ,n}=\{x\in P_{\xi }:\mathrm{St}(x,{\mathcal{H}}_n)\subseteq U_{\xi }\}\text{mbox}and{F}_{\xi ,n}=\overline{D_{\xi ,n}}.$$

$$P_{\xi }=\bigcup _{n\in \omega }{F}_{\xi ,n}$$

$$\phi (\xi )=\{k\in \omega :F_k\subseteq P_{\xi }\}.$$

Full text

This result generalizes one theorem of Rene Baire on classification of barely continuous functions.

1. Introduction

A map $f:X\to Y$ between topological spaces $X$ and $Y$ is said to be

Baire-one, if it is a pointwise limit of a sequence of continuous maps $f_{n}:X\to Y$;

• -

(functionally) $F_{\sigma}$-measurable or of the first (functional) Borel class, if the preimage $f^{-1}(V)$ of any open set $V\subseteq Y$ 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 $F\subseteq X$ has a point of continuity.

Let us observe that the term ”barely continuous” belongs to Stephens [16]. However, barely continuous 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 theorem is well-known [2].

Theorem A**.**

For a complete metric space $X$ and a function $f:X\to\mathbb{R}$ the following conditions are equivalent:

1. (1)

$f$* is Baire-one;* 2. (2)

$f$* is $F_{\sigma}$-measurable;* 3. (3)

$f$* is barely continuous.*

Recall that a map $f:X\to Y$ between topological space $X$ and a metric space $Y$ is said to be fragmented, if for all $\varepsilon>0$ and nonempty closed set $F\subseteq X$ there exists a relatively open set $U\subseteq F$ such that ${\rm diam}f(U)<\varepsilon$. The above notion was supposed by Jayne and Rogers [6] in connection with 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 $f:\mathbb{Q}\to\mathbb{R}$, $f(r_{n})=1/n$, where $\mathbb{Q}=\{r_{n}:n\in\mathbb{N}\}$ is the set of all rational numbers. Notice that $f$ is fragmented and everywhere discontinuous.

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 $f:X\to Y$. The following conditions are equivalent:

1. (i)

$f$* is $F_{\sigma}$-measurable and $\sigma$-discrete;* 2. (ii)

$f$* is fragmented.*

Moreover, if $Y$ is a convex subset of a Banach space, they are equivalent to:

1. (iii)

$f$* is Baire-one.*

Let us observe that a similar result for $Y=\mathbb{R}$ 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-connected space and $f:X\to Y$ is fragmented, then $f\in{\rm B}_{1}(X,Y)$.

The aim of this note is to extend the above mentioned results on maps defined on paracompact spaces which are not necessarily perfect (recall that a topological space is perfect if every its open subset is $F_{\sigma}$).

The convenient tool of investigation of fragmented functions on non-perfect spaces is a concept of 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 notion of $\sigma$-discrete decomposability. As an application of this result we obtain (Theorem 3) that for a paracompact space $X$, a metric space $Y$ and a functionally fragmented map $f:X\to Y$ the following propositions hold: (i) $f$ is $\sigma$-discrete and functionally $F_{\sigma}$-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 Lindelöff.

2. Preliminaries

Let $\mathscr{U}=(U_{\xi}:\xi\in[0,\alpha])$ be a transfinite sequence of subsets of a topological space $X$. Then $\mathscr{U}$ is regular in $X$, if

1. (a)

each $U_{\xi}$ is open in $X$; 2. (b)

$U_{0}=\emptyset$, $U_{\alpha}=X$ and $U_{\xi}\subseteq U_{\eta}$ for all $0\leq\xi\leq\eta<\alpha$; 3. (c)

$U_{\gamma}=\bigcup_{\xi<\gamma}U_{\xi}$ for every limit ordinal $\gamma\in[0,\alpha]$.

Proposition 1**.**

[12, Proposition 1]** Let $X$ be a topological space, $(Y,d)$ be a metric space and $\varepsilon>0$. For a map $f:X\to Y$ the following conditions are equivalent:

1. (1)

$f$* is $\varepsilon$-fragmented;* 2. (2)

there exists a regular sequence $\mathscr{U}=(U_{\xi}:\xi\in[0,\alpha])$ in $X$ such that ${\rm diam}f(U_{\xi+1}\setminus U_{\xi})<\varepsilon$ for all $\xi\in[0,\alpha)$.

If a sequence $\mathscr{U}$ satisfies condition (2) of the previous proposition, then it is called $\varepsilon$-associated with $f$ and is denoted by $\mathscr{U}_{\varepsilon}(f)$.

We say that an $\varepsilon$-fragmented map $f:X\to Y$ is functionally $\varepsilon$-fragmented if $\mathscr{U}_{\varepsilon}(f)$ can be chosen such that every set $U_{\xi}$ is functionally open in $X$. Further, $f$ is functionally fragmented, if it is functionally $\varepsilon$-fragmented for each $\varepsilon>0$.

A map $f$ is (functionally) countably fragmented, if $f$ is (functionally) fragmented and every sequence $\mathscr{U}_{\varepsilon}$ can be chosen to be countable.

3. A Lemma

Let $\mathscr{A}$ be a family of subsets of a topological space $X$. Then $\mathscr{A}$ is called

• •

discrete, if each point $x\in X$ has a neighborhood which intersects at most one set from $\mathscr{A}$;

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strongly functionally discrete or, briefly, sfd-family, if there exists a discrete family $(U_{A}:A\in\mathscr{A})$ of functionally open subsets of $X$ such that $\overline{A}\subseteq U_{A}$ for every $A\in\mathscr{A}$.

Let us observe that every discrete family is strongly functionally discrete in collectionwise normal space.

Lemma 2**.**

(cf. [3, Theorem 2])* Let $\mathscr{U}=(U_{\xi}:\xi\in[0,\alpha])$ be a regular family of functionally open sets in a paracompact space $X$. Then there exists a sequence $(\mathscr{F}_{n})_{n\in\omega}$ of families $\mathscr{F}_{n}=(F_{\xi,n}:\xi\in[0,\alpha])$ such that*

1. (1)

$U_{\xi}\setminus\bigcup_{\eta<\xi}U_{\eta}=\bigcup_{n\in\omega}F_{\xi,n}$* for all $\xi\in[0,\alpha)$,* 2. (2)

$\mathscr{F}_{n}$* is an sfd-family in $X$ for all $n\in\omega$,* 3. (3)

$F_{\xi,n}$* is closed in $X$ for all $n\in\omega$ and $\xi\in[0,\alpha)$.*

Proof.

For every $\xi\in[1,\alpha]$ we denote $P_{\xi}=U_{\xi}\setminus\bigcup_{\eta<\xi}U_{\eta}$. Since every $P_{\xi}$ is functionally $G_{\delta}$ in $X$ as a difference of two functionally open sets, we can choose a sequence $(G_{\xi,n})_{n\in\omega}$ of functionally open sets such that

[TABLE]

We put

[TABLE]

and define by the induction on $k$ sequences $(\mathscr{U}_{i}:i\in I)$ and $(\mathscr{V}_{i}:i\in I)$ of open coverings of $X$ such that

1. (a)

$\mathscr{U}_{\emptyset}=\mathscr{U}$; 2. (b)

$\mathscr{V}_{i}$ is a locally finite barycentric refinement of $\mathscr{U}_{i}$ for all $i\in\omega^{k}$; 3. (c)

for all $i\in\omega^{k}$ and $n\in\omega$ we have $\mathscr{U}_{(i,n)}=(U_{\xi,(i,n)}:\xi\in[0,\alpha))$, where

[TABLE]

for all $k\in\omega$. Let us observe that the existence of families $\mathscr{V}_{i}$ follows from the paracompactness of $X$ (see [4, Theorem 5.1.12]).

Notice that

[TABLE]

because $\mathscr{V}_{i}$ is an open covering of $X$. Therefore, since $(P_{\xi}:\xi\in[0,\alpha])$ is a partition of $X$, $\mathscr{U}_{(i,n)}$ defined in (c) covers $X$ for all $n\in\omega$.

For every $x\in X$ we put

[TABLE]

and show that

[TABLE]

Assume to the contrary that there exists $x\in X$ such that (3.1) is not true. Since each family $\mathscr{V}_{i}$ is locally finite refinement of $\mathscr{U}$, for every $i\in I$ there is $\xi_{i}$ such that ${\rm St}(x,\mathscr{V}_{i})\subseteq U_{\xi_{i}}$. Let $\xi(x)=\min\{\xi_{i}:i\in I\}$. Then $\xi(x)>\mu(x)$. Therefore, $x\not\in P_{\xi(x)}$ and we can take $j\in\omega$ such that $x\not\in G_{\xi(x),j}$.

From the definition of the sequence $\mathscr{U}_{(i,j)}$ it follows that $x\not\in U_{\xi(x),(i,j)}$. Since ${\rm St}(x,\mathscr{V}_{i})\subseteq U_{\xi(x)}$, we have $x\not\in U_{\xi,(i,j)}$ for all $\xi>\xi(x)$. Therefore,

[TABLE]

By (b) there exists $\beta\in[0,\alpha)$ such that ${\rm St}(x,\mathscr{V}_{(i,j)})\subseteq U_{\beta,(i,j)}$. It follows from (3.2) that $\beta<\xi(x)$. The inclusion $U_{\beta,(i,j)}\subseteq U_{\beta}$ contradicts to the choice of $\xi(x)$.

Let $(\mathscr{V}_{i}:i\in I)=(\mathscr{H}_{n}:n\in\omega)$. Now for all $\xi\in[0,\alpha)$ and $n\in\omega$ we put

[TABLE]

We will show that

[TABLE]

for all $\xi\in[0,\alpha)$. Property (3.1) implies that $P_{\xi}\subseteq\bigcup_{n\in\omega}F_{\xi,n}$. Now assume that $x\in F_{\xi,n}$ for some $\xi$ and $n$. Put $O={\rm St}(x,\mathscr{H}_{n})\cap U_{\mu(x)}$. Then $O\cap D_{\xi,n}\neq\emptyset$. Take any $y\in O$. Since $y\in U_{\mu(x)}$ and $y\in P_{\xi}$, $\mu(x)\geq\xi$. The inclusions ${\rm St}(y,\mathscr{H}_{n})\subseteq U_{\xi}$ and $y\in{\rm St}(x,\mathscr{H}_{n})$ imply that $x\in U_{\xi}$. Hence, $\mu(x)\leq\xi$. Therefore, $\mu(x)=\xi$. Then $x\in P_{\xi}$. Moreover, it follows that the family $\mathscr{F}_{n}=(F_{\xi,n}:\xi\in[0,\alpha))$ is discrete in $X$.

Since $X$ is paracompact, $X$ is collectionwise normal, which implies that $\mathscr{F}_{n}$ is strongly functionally discrete family for all $n\in\omega$. ∎

4. 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 any two continuous maps $f,g:X\to Y$ there exists a continuous map $h:X\to Y$ such that $h|_{A}=f|_{A}$ and $h|_{B}=g|_{B}$;

• •

a weak adhesive for $X$, if for any two points $y,z\in Y$ and disjoint functionally closed sets $A$ and $B$ in $X$ there exists a continuous map $h:X\to Y$ such that $h|_{A}=y$ i $h|_{B}=z$;

• •

a locally weak adhesive for $X$, if for every $y\in Y$ and every neighborhood $V\subseteq Y$ of $y$ there exists a neighborhood $U$ of $y$ such that $U\subseteq V$ and for every $z\in U$ there exists a continuous map $h:X\to V$ 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 strongly zero dimensional space $X$; a path-connected space $Y$ is an adhesive for any compact space $X$ each point 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-connected space is a (locally) weak adhesive for any $X$.

A family $\mathscr{B}$ of subsets of a topological space $X$ is said to be a base for a map $f:X\to Y$, if for every open set $V\subseteq Y$ there exists a subfamily $\mathscr{B}_{V}$ of $\mathscr{B}$ such that $f^{-1}(V)=\bigcup_{B\in\mathscr{B}_{V}}B$.

A map $f:X\to Y$ is $\sigma$-discrete, if there is a sequence $(\mathscr{B}_{n})_{n\in\omega}$ of discrete families of sets in $X$ such that the family $\bigcup_{n\in\omega}\mathscr{B}_{n}$ is a base for $f$.

Theorem 3**.**

Let $X$ be a paracompact space, $Y$ be a metric space and $f:X\to Y$ be a functionally fragmented map. Then

1. (1)

$f$* is $\sigma$-discrete and functionally $F_{\sigma}$-measurable;* 2. (2)

$f$* is a Baire-one function, if $Y$ is weak adhesive and weak locally adhesive for $X$;* 3. (3)

$f$* is countably functionally fragmented, if $X$ is Lindelöff.*

Proof.

1) For every $n\in\mathbb{N}$ we choose a family $\mathscr{U}_{1/n}(f)=(U_{\xi,n}:\xi\in[0,\alpha_{n}])$ consisting of functionally open sets $U_{\xi,n}$. We claim that the family $\mathscr{P}=\bigcup_{n\in\mathbb{N}}\mathscr{P}_{n}$ is a base for $f$, where $\mathscr{P}_{n}=(U_{\xi,n}\setminus\bigcup_{\eta<\xi}U_{\eta,n}:\xi\in[0,\alpha_{n}])$, $n\in\mathbb{N}$. Indeed, fix an open set $V$ in $Y$ and take any $x\in f^{-1}(V)$. Find $n\in\mathbb{N}$ such that an open ball $B$ with the center at $f(x)$ and radius $1/n$ contains in $V$. Since $\mathscr{P}_{n}$ is a partition of $X$, there exists $\xi\in[0,\alpha_{n}]$ such that $x\in P_{\xi,n}=U_{\xi,n}\setminus\bigcup_{\eta<\xi}U_{\eta,n}$. Evidently, $f(P_{\xi,n})\subseteq B\subseteq V$.

By Lemma 2 for every $n\in\mathbb{N}$ there exists a sequence $(\mathscr{F}_{n,k})_{k\in\omega}$ of families $\mathscr{F}_{n,k}=(F_{\xi,n,k}:\xi\in[0,\alpha_{n}])$ which satisfies conditions (1)–(3) of Lemma 2. Properties (1) and (2) imply that the family $\mathscr{B}=\bigcup\limits_{k,n}\mathscr{F}_{n,k}$ is a $\sigma$-discrete base for $f$ consisting of closed sets. It follows that $f$ is $F_{\sigma}$-measurable and a $\sigma$-discrete map. Finally, [7, Proposition 2.6 (iv)] implies that $f$ is functionally $F_{\sigma}$-measurable.

Property 2) follows from 1) and [8, Theorem 3.2].

3) It is enough to show that every regular sequence consisting of functionally open sets in a Lindelöff space $X$ is countable.

Let $\mathscr{U}=(U_{\xi}:\xi\in[0,\alpha])$ be a regular covering of $X$ by functionally open sets $U_{\xi}$. There exists a sequence $(\mathscr{F}_{n})_{n\in\omega}$ of families in $X$ such that conditions (1)–(3) of Lemma 2 are valid. Notice that every family $\mathscr{F}_{n}$ is at most countable, since it is discrete and $X$ is Lindelöff. We consider an enumeration $\{F_{k}:k\in\omega\}$ of the family $\bigcup_{n\in\omega}\mathscr{F}_{n}$. Let $\varphi:[0,\alpha)\to 2^{\omega}$ be a map,

[TABLE]

Since $(\varphi(\xi):\xi\in[0,\omega_{1}))$ is a family of mutually disjoint subsets of $\omega$, it is at most countable. ∎

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 paracompact Hausdorff space is functionally fragmented?