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Extension of $b_f$-continuous functions defined on a product of $b_f$-groups

Last modified: 01-10-2019

#### Abstract

From now on, $X$ will stand for a Tychonoff space.

A subset $B$ of a space $X$ is said to be \emph{bounded} (in $X$) if every

real-valued continuous function on $X$ is bounded on $B$ or, equivalently, every locally finite sequence $\left\{U\sb{n} : n\in\mathbb{B}\right\}$ of pairwise disjoint open sets meeting $B$ is finite. Spaces

which are

bounded in themselves are called \emph{pseudocompact}. Given a space $X$, the family of all bounded subsets of $X$ is denoted by $\beta$. If $\alpha$ is a cover of a space $X$, we say that a function $f$

from a space $X$ into a space $Y$ is $\alpha\sb{f}$-continuous if the restriction of

$f$ to each member of $\alpha$ can be extended to a continuous function

on the whole $X$. A space $X$ is called an $\alpha\sb{f}$\emph{-space} if every

real-valued $\alpha\sb{f}$-continuous function on $X$ is continuous. By a $b_f$-group it is understood a topological groups whose underlying topological space is a $b_f$-space.

Locally

pseudocompact spaces and

$k\sb{r}$-spaces (spaces $X$ where a real-valued function is continuous whenever

its restriction to each compact subset of $X$ is continuous) are examples of $\beta\sb{f}$-spaces.

Thus, locally

compact spaces, first countable spaces (in particular, metrizable spaces) are

$\beta\sb{f}$-spaces too.

The theory of $z$-closed projections, the distribution of the

functor of the

Dieudonn\'e completion, compactness of functions spaces in the

topology of the

pointwise convergence, and locally pseudocompact groups

are some of

the frameworks where $\beta\sb{f}$-spaces arise in a natural way.

Let $F(X)$ denote the set of all real-valued functions from a set $X$.

We denote by $\tau\sb{\beta}$ the topology of uniform convergence on

members of $\beta$. It is a well-known fact that $((F(X),\tau\sb{\beta})$

is a Tychonoff space. Indeed, the family of all subsets of $F(X)\times

F(X)$ of the form\begin{small}

\[

U(A,\varepsilon)=\left\{(f,g)\in F(X)\times

(F(X) : \sup\sb{a\in

A} \left |f(a),g(a)\right |<\varepsilon\right\},

\]\end{small}

for all $A\in \beta$ and all $\varepsilon>0$, is a subbase for a

(Hausdorff) uniformity $\mathcal{U}\sb{\beta}$ on $F(X)$ which induces the topology $\tau\sb{\beta}$.

We present some results concerning extensions of $b_f$-continuous functions defined on a product of $b_f$-groups. Let $\mu$ denote the exponential map from $F(X\times Y)$ into $F(X,F\sb{\tau_\beta}(Y))$ and, as usual, $\beta (X)$ (respectively, $C(X)$) denotes the Stone-\v{C}ech compactification of a space $X$ (respectively, the ring of all real-valued continuous functions on $X$) . The results are similar in flavor to the following theorem.

\begin{theorem}

Let $G$ and $H$ be two $b_f$-groups. If $f$ is a $b_f$-continuous function on $G\times H$, then the following conditions are equivalent:

\begin{itemize}\itemsep=-0.1cm

\item[(a)] $f$ can be extended to a $b_f$-continuous function on $\beta (G)\times H$; \medskip

\item[(b)] $f$ can be extended to a $b_f$-continuous function on $G\times \beta (H)$;

\item[(c)] $f$ can be extended to a $b_f$-continuous function on $\beta (G)\times \beta (H)$;

\item[(d)] the closure of $\mu (G)$ in $C\sb{\tau_\beta}(H)$ is compact;

\item[(e)] $\mu (G)$ is a compact subspace of $C\sb{\tau_\beta}(H)$.

\end{itemize}

\end{theorem}\vspace*{-0.3cm}

Our results apply in several ways. For example, to get a characterization of when the product of two $b_f$-groups is a $b_f$-group.

A subset $B$ of a space $X$ is said to be \emph{bounded} (in $X$) if every

real-valued continuous function on $X$ is bounded on $B$ or, equivalently, every locally finite sequence $\left\{U\sb{n} : n\in\mathbb{B}\right\}$ of pairwise disjoint open sets meeting $B$ is finite. Spaces

which are

bounded in themselves are called \emph{pseudocompact}. Given a space $X$, the family of all bounded subsets of $X$ is denoted by $\beta$. If $\alpha$ is a cover of a space $X$, we say that a function $f$

from a space $X$ into a space $Y$ is $\alpha\sb{f}$-continuous if the restriction of

$f$ to each member of $\alpha$ can be extended to a continuous function

on the whole $X$. A space $X$ is called an $\alpha\sb{f}$\emph{-space} if every

real-valued $\alpha\sb{f}$-continuous function on $X$ is continuous. By a $b_f$-group it is understood a topological groups whose underlying topological space is a $b_f$-space.

Locally

pseudocompact spaces and

$k\sb{r}$-spaces (spaces $X$ where a real-valued function is continuous whenever

its restriction to each compact subset of $X$ is continuous) are examples of $\beta\sb{f}$-spaces.

Thus, locally

compact spaces, first countable spaces (in particular, metrizable spaces) are

$\beta\sb{f}$-spaces too.

The theory of $z$-closed projections, the distribution of the

functor of the

Dieudonn\'e completion, compactness of functions spaces in the

topology of the

pointwise convergence, and locally pseudocompact groups

are some of

the frameworks where $\beta\sb{f}$-spaces arise in a natural way.

Let $F(X)$ denote the set of all real-valued functions from a set $X$.

We denote by $\tau\sb{\beta}$ the topology of uniform convergence on

members of $\beta$. It is a well-known fact that $((F(X),\tau\sb{\beta})$

is a Tychonoff space. Indeed, the family of all subsets of $F(X)\times

F(X)$ of the form\begin{small}

\[

U(A,\varepsilon)=\left\{(f,g)\in F(X)\times

(F(X) : \sup\sb{a\in

A} \left |f(a),g(a)\right |<\varepsilon\right\},

\]\end{small}

for all $A\in \beta$ and all $\varepsilon>0$, is a subbase for a

(Hausdorff) uniformity $\mathcal{U}\sb{\beta}$ on $F(X)$ which induces the topology $\tau\sb{\beta}$.

We present some results concerning extensions of $b_f$-continuous functions defined on a product of $b_f$-groups. Let $\mu$ denote the exponential map from $F(X\times Y)$ into $F(X,F\sb{\tau_\beta}(Y))$ and, as usual, $\beta (X)$ (respectively, $C(X)$) denotes the Stone-\v{C}ech compactification of a space $X$ (respectively, the ring of all real-valued continuous functions on $X$) . The results are similar in flavor to the following theorem.

\begin{theorem}

Let $G$ and $H$ be two $b_f$-groups. If $f$ is a $b_f$-continuous function on $G\times H$, then the following conditions are equivalent:

\begin{itemize}\itemsep=-0.1cm

\item[(a)] $f$ can be extended to a $b_f$-continuous function on $\beta (G)\times H$; \medskip

\item[(b)] $f$ can be extended to a $b_f$-continuous function on $G\times \beta (H)$;

\item[(c)] $f$ can be extended to a $b_f$-continuous function on $\beta (G)\times \beta (H)$;

\item[(d)] the closure of $\mu (G)$ in $C\sb{\tau_\beta}(H)$ is compact;

\item[(e)] $\mu (G)$ is a compact subspace of $C\sb{\tau_\beta}(H)$.

\end{itemize}

\end{theorem}\vspace*{-0.3cm}

Our results apply in several ways. For example, to get a characterization of when the product of two $b_f$-groups is a $b_f$-group.

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