On the Quasinormal Convergence of Functions

Abstract

In the paper, it is proved that a topological space $X$ is a $QN$-space if and only if every image of the space $X$ under a Baire mapping to the Baire space $\mathbb{N}^{\mathbb{N}}$ is bounded. It is shown that there exists a compact $QN$-space such that its image under a Borel mapping to the Baire space $\mathbb{N}^{\mathbb{N}}$ is unbounded. The existence of such a space answers a question of L. Bukovský and J. Haleš. Generalizations of results of N. N. Kholshchevnikova concerning the representation of functions on subsets of the number line by trigonometric series are obtained.

Introduction

Let $X$ be a topological space, and let $f$ and $f_n$ be functions from $X$ to $\mathbb{R}$, where $\mathbb{R}$ is the set of real numbers. A sequence $\{f_n\}$ of functions converges quasinormally (briefly, $QN$-converges) to $f$ on $X$ if there is a sequence $\{\epsilon_n\}$ of positive numbers with $\lim_{n\to \infty}\epsilon_n=0$ such that, for every $x\in X$, there is an index $n_x$ such that $|f(x)-f_n(x)|<\epsilon_n$ for all $n\ge n_x$. There is a simple criterion for quasinormal convergence: a sequence $\{f_n\}$ of functions converges quasinormally to a function $f$ on $X$ if and only if $X$ is a countable union of sets on each of which the sequence $\{f_n\}$ converges uniformly ([1], [2]). Quasinormal convergence was considered, for example, when constructing $R$- and $N$-sets for trigonometric series [3, p. 737 (Russian transl.)] to represent a real-valued function defined on a subset of the number line $\mathbb{R}$ by trigonometric series [4], and also in the set of Borel functions defined over a perfectly normal topological space [5]–[7]. Note that every convergent sequence on a countable set is $QN$-convergent, which fails to hold on a set of the cardinality of the continuum. Under the assumption of the negation of the continuum hypothesis, which is consistent with $ZFC$, small cardinals occur; this term is used for the intermediate cardinals between the countable cardinal and the cardinality of the continuum. Examples of small cardinals are $\mathfrak {p}$ and $\mathfrak{b}$. Recall that $\mathfrak{p}$ is the least cardinal number such that, if a family $\xi$ of subsets of the set $\mathbb{N}$ of positive integers is of cardinality less than $\mathfrak{p}$ and if the intersection of any finite subfamily of $\xi$ is infinite, then there is an infinite subset $B$ of $\mathbb{N}$ such that the set $B\setminus A$ is finite for every $A\in\xi$.

For $f,g\in \mathbb{N}^{\mathbb{N}}$, we write $f\le^{*} g$ if $f(n)\le g(n)$ for all $n$ except for finitely many of them. A set $B\subset \mathbb{N}^{\mathbb{N}}$ is unbounded if the set of all increasing enumerations of $B$ is unbounded in $\mathbb{N}^{\mathbb{N}}$ with respect to $\le^{*}$; $\mathfrak{b}$ is the minimum cardinality of a $\le^{*}$-unbounded subset of $\mathbb{N}^{\mathbb{N}}$. Note that, in every model of set theory (consistent with $ZFC$), $\omega_1 \le \mathfrak{p}\le \mathfrak{b}\le \mathfrak{c}$. If we assume that Martin’s axiom holds, then $\mathfrak{p}=\mathfrak{b}= \mathfrak{c}$.

Bukovský, Reclaw, and Repický called a topological space $X$ a $QN$-space if, for every sequence $\{f_n\}$ of continuous real-valued functions pointwise convergent to the zero function $\boldsymbol{0}$ on $X$, the sequence $\{f_n\}$ quasinormally converges to $\boldsymbol{0}$ on $X$. As proved in [5], every perfectly normal space of cardinality less than $\mathfrak{b}$ is a $QN$-space. An example of a space of cardinality $\mathfrak{b}$ which is not a $QN$-space was constructed ibidem.

We show that an arbitrary topological space $X$ is a $QN$-space if and only if the image of $X$ under any Baire mapping to the Baire space $\mathbb{N}^{\mathbb{N}}$ is bounded. In particular, for an arbitrary Baire function $f$ defined on a topological space $X$, where $|X|<\mathfrak{b}$, and a sequence of Baire functions $\{f_n\}$ which converges pointwise to $f$ on $X$, it follows that $\{f_n\}$ quasinormally converges to $f$ on $X$. Our result (for the class of arbitrary topological spaces) generalizes the result of Bukovský, Reclaw, and Repický (obtained for the class of perfectly normal topological spaces). Since there are models of set theory for which $\mathfrak{p}<\mathfrak{b}$, we obtain a generalization of the following result of N. N. Kholshchevnikova: for an arbitrary function $f$ defined on $A\subset \mathbb{R}$ , where $|A|<\mathfrak{p}$, there is a sequence of continuous functions on $\mathbb{R}$ which converges quasinormally to $f$ on $A$. Since every Borel function defined on a perfectly normal space $X$ is a Baire function, it follows that our result generalizes that of Tsaban and Zdomskyy: a perfectly normal space $X$ is a $QN$-space if and only if every Borel image of the space $X$ in the Baire space $\mathbb{N}^{\mathbb{N}}$ is bounded [8]. We construct an example of a compact (not perfectly normal) space $X$ such that $X$ is a $QN$-space but admits a unbounded Borel mapping to the Baire space $\mathbb{N}^{\mathbb{N}}$ such that the image of space $X$ is unbounded. The existence of such an example solves an open problem of Bukovský and Haleš in [6, Problem 22].

Arhangel’skii introduced the $\alpha_1$ property for an arbitrary topological space [9]. A space $X$ has the property $\alpha_1$ (is an $\alpha_1$-space) if, for every point $x\in X$ and any countable family of sequences $A_n$ converging to $x$, there is a sequence $B$ converging to $x$ and such that $A_n\setminus B$ is finite for every $n\in \mathbb{N}$.

For an arbitrary topological space $X$, $C_p(X)$ denotes the family of all continuous real-valued functions on $X$ regarded as a subspace of the Tychonoff product $\mathbb{R}^X$, and $C_p(X,\{0,1\})$ is the set of all continuous mappings from $X$ to the discrete two-point space $\{0,1\}$ with the topology of pointwise convergence. As shown in [6] and [7], $X$ is a $QN$-space if and only if $C_p(X)$ has the property $\alpha_1$.

Recall that a mapping from a topological space $X$ to a topological space $Y$ is said to be Borel (Baire) if the preimage of every open set is a Borel (Baire) subset of the space $X$.

In the paper we identify the Cantor space $\{0,1\}^{\mathbb{N}}$ with the family $\mathcal{P}(\mathbb{N})$ of all subsets of the set $\mathbb{N}$ of positive integers. The family of infinite subsets of $\mathbb{N}$ is denoted by $[\mathbb{N}]^{\infty}$. We also identify every element $a\in [\mathbb{N}]^{\infty}$ with its increasing enumeration, which is an element of the Baire space $\mathbb{N}^\mathbb{N}$. For every positive integer $n$, $a(n)$ stands for the $n$th element in the increasing enumeration of the element $a$. Thus, $[\mathbb{N}]^{\infty}\subseteq \mathbb{N}^\mathbb{N}$, and the topology on the space $[\mathbb{N}]^{\infty}$ (a subspace of the space $\mathcal{P}(\mathbb{N})$) coincides with the subspace topology induced from the space $\mathbb{N}^\mathbb{N}$.

By a zero set we mean a subset of the space $X$ which can be represented as the preimage of zero under a continuous real-valued function. The complement to a zero set is called a cozero set in the space $X$.

We use the terminology and notation adopted in [5], [8], [10], and [11].

1. Main Results

A cover $\mathcal{U}$ of a topological space $X$ is called a $\gamma$-cover if $\mathcal{U}$ is infinite and every point of the space $X$ belongs to all elements of the cover $\mathcal{U}$ except for finitely many of them. Note that every infinite subset of $\mathcal{U}$ is also a $\gamma$-cover of the space $X$. We assume below that all $\gamma$-covers are countable. For a cover $\mathcal{U}$, a subset $\mathcal{V}\subseteq \mathcal{U}$ is said to be cofinite if $\mathcal{V}$ coincides with $\mathcal{U}$ except for finitely many elements.

The following result was proved by Bukovský and Haleš [6, Theorem 17] and, independently, by Sakai in [7, Theorem 3.7].

Lemma 1.

The following conditions are equivalent :

• (1) $C_p(X,\{0,1\})$ is an $\alpha_1$ -space ;

• (2) for every family $\{\mathcal{U}_n:n\in \mathbb{N}\}$ of pairwise disjoint clopen (open-and-closed ) $\gamma$ -covers of the space $X$ there are cofinite $\mathcal{V}_n\subseteq \mathcal{U}_n$ , $n\in \mathbb{N}$ , such that $\bigcup_n\mathcal{V}_n$ is a $\gamma$ -cover of the space $X$ .

Every bijectively enumerated family $\mathcal{U}=\{U_n : n\in \mathbb{N}\}$ of subsets of the set $X$ induces the Marczewski mapping $\mathbb{U}\colon X \to \mathcal{P}(\mathbb{N})$ defined by the rule $\mathbb{U}(x)=\{n\in \mathbb{N}: x\in U_n\}$ for every $x\in X$.

A function $f$ defined on a space $X$ is the discrete limit of functions $f_n$, $n\in \mathbb{N}$, if, for every $x\in X$, $f_n(x)=f(x)$ for all $n$ except for finitely many of them.

The following lemma is proved similarly to Lemma 11 of [8].

Lemma 2.

Let $C_p(X)$ be an $\alpha_1$ -space, and let $\mathcal{U}=\{U_n \colon n\in \mathbb{N}\}$ be a bijectively enumerated family of functionally open subsets of the space $X$ . Then the Marczewski mapping $\mathbb{U}\colon X \to \mathcal{P}(\mathbb{N})$ is a discrete limit of continuous functions.

Proof.

Since $C_p(X)$ is an $\alpha_1$-space, it follows that the space $X$ has the property that, for any two disjoint zero sets $F_1$ and $F_2$, there exists a clopen set $W$ such that $F_1\subseteq W$ and $W\cap F_2=\varnothing$, i.e., the space $X$ has the property $\operatorname{Ind}_ZX=0$ [7]. Since, for every $n$, the set $U_n$ is cozero, it follows that $U_n$ can be represented in the form $\bigcup_m W^n_m$, where $\{W^n_m: m\in \mathbb{N}\}$ is a disjoint family of clopen sets. We may assume that the families $\{W^n_m: m\in \mathbb{N}\}$ are pairwise disjoint for distinct $n$.

Thus, the families $\mathcal{U}_n=\{X\setminus W^n_m : m\in \mathbb{N}\}$ are disjoint clopen $\gamma$-covers of the space $X$. By Lemma 1, there are $k_n$, $n\in \mathbb{N}$, and subsets $\mathcal{V}_n=\{X\setminus W^n_m : m\ge k_n\} \subseteq \mathcal{U}_n$, $n\in \mathbb{N}$, such that $\bigcup \mathcal{V}_n$ is a $\gamma$-cover of the space $X$. This implies that $\mathcal{V}=\{\bigcap_{m=k_n}^{\infty} X \setminus W^n_m : n\in \mathbb{N}\}$ is a $\gamma$-cover of the space $X$.

Let $U^n_m=\bigcup_{i=1}^{\max\{m,k_n\}} W_i^n$ for any $n$ and $m$. For every $m$, we define $\Psi_m\colon X \to \mathcal{P}(\mathbb{N})$ by the rule $\Psi_m(x)=\{n : x\in U^n_m\}$. Since every set $U^n_m$ is clopen, it follows that the mapping $\Psi_m(x)$ is continuous. It remains to prove that the Marczewski mapping $\mathbb{U}\colon X\to \mathcal{P}(\mathbb{N})$ is the discrete limit of the mappings $\Psi_m(x)$, $m\in \mathbb{N}$.

Let $x\in X$. There is an $n_x\in \mathbb{N}$ such that $x\in \bigcup_{m=k_n}^{\infty} X\setminus W^n_m$ for all $n>n_x$. For every $n<n_x$ such that $x\in U_n$, let $m_n$ be the index for which $x\in U^n_{m_n}$. We set $M=\max\{m_n : n<n_x\}$.

Suppose that $m\ge M$. Let us show that $x\in U^n_m$ if and only if $x\in U_n$. The inclusion $U^n_m\subseteq U_n$ holds by construction. Let $x\in U_n$. Consider the following two cases:

• (1) $n<n_x$; in this case, $x\in U^m_n$, since $m\ge M \ge m_n$;

• (2) $n\ge n_x$; in this case, $x\in \bigcap_{i=k_n}^{\infty} X \setminus W^n_i$, and since $x\in U_n=\bigcup_m W^n_m$, it follows that

  $$x\in\bigcup_{i=1}^{k_n-1}W^n_i\subseteq U^n_m.$$

Thus, for every $x\in X$, there is an $M$ such that $\Psi_m(x)=\mathcal{U}(x)$ for all $m\ge M$. This completes the proof of the lemma.

Lemma 3.

Let $C_p(X)$ be an $\alpha_1$ -space, and let $\mathcal{F}=\{F_n : n\in \mathbb{N}\}$ be a bijectively enumerated family of zero sets of the space $X$ . Then the Marczewski mapping $\mathbb{G}\colon X \to \mathcal{P}(\mathbb{N})$ for the family $\mathcal{F}$ is a discrete limit of continuous functions.

Proof.

It suffices to consider the Marczewski mapping $\mathbb{U}\colon X \to \mathcal{P}(\mathbb{N})$ for the family

$$\mathcal{U}=\{X\setminus F_n : n\in \mathbb{N}\}$$

of cozero sets in the space $X$. By Lemma 2, the family of continuous mappings $\Psi_m\colon X\to\mathcal{P}(\mathbb{N})$ defined by the rule $\Psi_m(x)=\{n : x\in U^n_m\}$ converges discretely to the Marczewski mapping $\mathbb{U}$. Note that, beginning with some index $m$, we have $x\in U^n_m$ if and only if $x\in X\setminus F_n$. This implies that $\Phi_m(x)=\{n : x\notin U^n_m\}$ converges discretely to the Marczewski mapping $\mathbb{G}$ for the family $\mathcal{F}=\{F_n : n\in \mathbb{N}\}$.

Theorem 1.

Let $X$ be a topological space. The following assertions are equivalent :

• (1) $X$ is a $QN$ -space ;

• (2) every image of $X$ under a Baire mapping to Baire space $\mathbb{N}^{\mathbb{N}}$ is bounded.

Proof.

(1) $\Rightarrow$ (2). Let $\Psi(X)\subseteq\mathbb{N}^{\mathbb{N}}$, where $\Psi\colon X\to\mathbb{N}^{\mathbb{N}}$ is a Baire mapping of the space $X$ to $\mathbb{N}^{\mathbb{N}}$. For all $n,k\in \mathbb{N}$, we consider the nonempty sets

$$Z_{n,k}=\bigcup \{\Psi^{-1}(y):y[n]=k,\,y\in \Psi(X)\}.$$

By a theorem of [12] (or by Theorem 3.5 of [7]), every Baire subset of $X$ is a $Z_{\sigma}$-set, i.e., a countable union of zero sets in the space $X$. Thus, every set $Z_{n,k}$ is representable in the form $Z_{n,k}=\bigcup_m F^m_{n,k}$, where $F^m_{n,k}$ is a zero set in $X$ for every $m\in \mathbb{N}$. We set $X_m=\bigcup \{F^m_{n,k} : n,k\in \mathbb{N}\}$. By Theorem 4.1 of [5] (which holds for an arbitrary $QN$-space $X$ and an arbitrary $Z_{\sigma}$-subset of $X$), the set $X_m$ is a $QN$-space. Consider the Marczewski mapping $\mathcal{G}_m\colon X_m\to\mathcal{P}(\mathbb{N}\times \mathbb{N})$ for $\{F^m_{n,k} : n,k\in \mathbb{N}\}$ defined by the rule $\mathcal{G}_m(x)= \{(n,k)\in \mathbb{N}\times \mathbb{N}: x\in F^m_{n,k}\}$. By Lemma 3, $\mathcal{G}_m$ is a discrete limit of continuous functions. By Theorem 4.8 of [5], $\mathcal{G}_m(X_m)$ (as a subspace of $\mathbb{N}^\mathbb{N}$) is a $QN$-space. It remains to note that $\Psi(X)=\bigcup_m\mathcal{G}_m(X_m)$. By Theorem 1.2 of [5], $\Psi(X)$ is a $QN$-space and, by Corollary 2.2 of [5], $\Psi(X)$ is bounded.

(2) $\Rightarrow$ (1). Let $\{f_n\}$ be a sequence of continuous real-valued functions such that $\{f_n\}$ converge pointwise to the function $\boldsymbol{0}$. Consider the Baire mapping $\Psi\colon X \to \mathbb{N}^{\mathbb{N}}$, where

$$\Psi(x)(n)=\min\biggl\{k: |f_m(x)|< \frac1n\text{ for all }m> k\biggr\}.$$

Then $\Psi(X)$ is bounded by some function $\varphi \in \mathbb{N}^{\mathbb{N}}$. This implies that $\{f_n\}$ quasinormally converges to the function $\boldsymbol{0}$. This completes the proof of the theorem.

Theorem 1 implies the following corollaries.

Corollary 1.

Let $f$ be an arbitrary Baire function defined on a topological space $X$ with $|X|<\mathfrak{b}$ , and let $\{f_n\}$ be a sequence of Baire functions pointwise converging to $f$ on $X$ . Then $\{f_n\}$ quasinormally converges to $f$ on $X$ .

Similarly to Theorem 2 and Theorem 3 of [4], the following corollaries hold (in an arbitrary model of set theory consistent with $ZFC$).

Corollary 2.

If $f\colon A\to \mathbb{R}$ is a Baire function on $A$ , $0\in A\subset \mathbb{R}$ , and $|A|<\mathfrak{b}$ , then there exists a sequence $\{a_n\}$ of rational numbers and a strictly increasing sequence $\{n_\nu\}$ of nonnegative integers (with $n_0=0$ ) such that

$$f(x)=f(0)+\sum_{\nu=0}^{\infty} \biggl(\,\sum_{n=n_{\nu}+1}^{n_{\nu+1}} a_n x^n\biggr),$$

where the series converges quasinormally on the set $A$ .

Corollary 3.

If $f\colon A\to \mathbb{R}$ is a Baire function on $A$ , where $A\subset (-\pi,\pi)$ and $|A|<\mathfrak{b}$ , then there exist sequences $\{a_n\}$ and $\{b_n\}$ of rational numbers and a strictly increasing sequence $\{n_\nu\}$ of nonnegative integers (with $n_0=0$ ) such that

$$f(x)=\frac{a_0}{2}+\sum_{\nu=0}^{\infty} \biggl(\,\sum_{n=n_{\nu}+1}^{n_{\nu+1}}a_n\cos nx+b_n\sin nx\biggr),$$

where the series converges quasinormally on the set $A$ .

The following theorem answers the following question of Bukovský and Haleš [6, Problem 22]: Can a $QN$-space have an unbounded image in the Baire space $\mathbb{N}^{\mathbb{N}}$?

Theorem 2.

There exists a compact space $X$ of cardinality $\mathfrak{b}$ and a Borel mapping $X\to \mathbb{N}^{\mathbb{N}}$ such that $X$ is a $QN$ -space and the image of $X$ under this Borel mapping is unbounded.

Proof.

Consider the Alexandrov one-point compactification $X=D\cup \{a\}$ of a discrete set $D$ of cardinality $\mathfrak{b}$, where $a$ is the unique nonisolated point. Let $\Psi\colon X \to B$ be a bijective mapping of the space $X$ onto an arbitrary unbounded set $B\subset \mathbb{N}^{\mathbb{N}}$ of cardinality $\mathfrak{b}$. Note that $X$ is a $QN$-space [5, Theorem 1.2], and the Borel image $\Psi(X)=B$ is an unbounded set in $\mathbb{N}^{\mathbb{N}}$.