Polytechnic University of Valencia Congress, Workshop Applied Topological Structures WATS'17

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

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.

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