TY - UNPB

T1 - BASIC PROPERTIES OF X FOR WHICH SPACES Cp(X) ARE DISTINGUISHED

AU - Kakol, Jerzy

AU - Leiderman, Arkady

PY - 2021

Y1 - 2021

N2 - In our paper [18] we showed that a Tychonoff space $X$ is a
$\Delta$-space (in the sense of [20], [30]) if and only if the locally
convex space $C_{p}(X)$ is distinguished. Continuing this research, we
investigate whether the class $\Delta$ of $\Delta$-spaces is invariant
under the basic topological operations. We prove that if $X \in \Delta$
and $\varphi:X \to Y$ is a continuous surjection such that $\varphi(F)$
is an $F_{\sigma}$-set in $Y$ for every closed set $F \subset X$, then
also $Y\in \Delta$. As a consequence, if $X$ is a countable union of
closed subspaces $X_i$ such that each $X_i\in \Delta$, then also $X\in
\Delta$. In particular, $\sigma$-product of any family of scattered
Eberlein compact spaces is a $\Delta$-space and the product of a
$\Delta$-space with a countable space is a $\Delta$-space. Our results
give answers to several open problems posed in \cite{KL}. Let $T:C_p(X)
\longrightarrow C_p(Y)$ be a continuous linear surjection. We observe
that $T$ admits an extension to a linear continuous operator
$\widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a
$\Delta$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are
metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is. Making
use of obtained results, we provide a very short proof for the claim
that every compact $\Delta$-space has countable tightness. As a
consequence, under Proper Forcing Axiom (PFA) every compact
$\Delta$-space is sequential. In the article we pose a dozen open
questions.

AB - In our paper [18] we showed that a Tychonoff space $X$ is a
$\Delta$-space (in the sense of [20], [30]) if and only if the locally
convex space $C_{p}(X)$ is distinguished. Continuing this research, we
investigate whether the class $\Delta$ of $\Delta$-spaces is invariant
under the basic topological operations. We prove that if $X \in \Delta$
and $\varphi:X \to Y$ is a continuous surjection such that $\varphi(F)$
is an $F_{\sigma}$-set in $Y$ for every closed set $F \subset X$, then
also $Y\in \Delta$. As a consequence, if $X$ is a countable union of
closed subspaces $X_i$ such that each $X_i\in \Delta$, then also $X\in
\Delta$. In particular, $\sigma$-product of any family of scattered
Eberlein compact spaces is a $\Delta$-space and the product of a
$\Delta$-space with a countable space is a $\Delta$-space. Our results
give answers to several open problems posed in \cite{KL}. Let $T:C_p(X)
\longrightarrow C_p(Y)$ be a continuous linear surjection. We observe
that $T$ admits an extension to a linear continuous operator
$\widehat{T}$ from $R^X$ onto $R^Y$ and deduce that $Y$ is a
$\Delta$-space whenever $X$ is. Similarly, assuming that $X$ and $Y$ are
metrizable spaces, we show that $Y$ is a $Q$-set whenever $X$ is. Making
use of obtained results, we provide a very short proof for the claim
that every compact $\Delta$-space has countable tightness. As a
consequence, under Proper Forcing Axiom (PFA) every compact
$\Delta$-space is sequential. In the article we pose a dozen open
questions.

KW - Mathematics - General Topology

M3 - ???researchoutput.researchoutputtypes.workingpaper.preprint???

BT - BASIC PROPERTIES OF X FOR WHICH SPACES Cp(X) ARE DISTINGUISHED

PB - arXiv:2104.10506 [math.GN]

ER -