Topological properties of some function spaces

DOI10.1016/J.TOPOL.2020.107248arXiv2004.05321MaRDI QIDQ6338524

Saak S. Gabriyelyan, Alexander V. Osipov

Publication date: 11 April 2020

Abstract: Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{mathcalI}$-Tychonoff space for some ideal $mathcalI$ of compact sets of $X$. Denote by $C_{mathcalI}(X,Y)$ the space of continuous functions from $X$ to $Y$ endowed with the $mathcalI$-open topology. We prove that $C_{mathcalI}(X,Y)$ is Fr'{e}chet - Urysohn iff $X$ has the property $gamm{a}_{mathcalI}$. We characterize zero - dimensional Tychonoff spaces $X$ for which the space $$ is sequential. Extending the classical theorems of Gerlits, Nagy and Pytkeev we show that if $Y$ is not compact, then $C_p(X,Y)$ is Fr'{e}chet - Urysohn iff it is sequential iff it is a $k$-space iff $X$ has the property $gamma$. An analogous result is obtained for the space of bounded continuous functions taking values in a metrizable locally convex space. Denote by $B_1(X,Y)$ and $B(X,Y)$ the space of Baire one functions and the space of all Baire functions from $X$ to $Y$, respectively. If $H$ is a subspace of $B(X,Y)$ containing $B_1(X,Y)$, then $H$ is metrizable iff it is a $sigma$ - space iff it has countable $c{s}^{*}$ - character iff $X$ is countable. If additionally $Y$ is not compact, then $H$ is Fr'{e}chet - Urysohn iff it is sequential iff it is a $k$ - space iff it has countable tightness iff $X_{alep{h}_0}$ has the property $gamma$, where $X_{alep{h}_0}$ is the space $X$ with the Baire topology. We show that if $X$ is a Polish space, then the space $B_1(X,mathbbR)$ is normal iff $X$ is countable.

Mathematics Subject Classification ID

Function spaces in general topology (54C35) General theory of locally convex spaces (46A03) Barrelled spaces, bornological spaces (46A08)

This page was built for publication: Topological properties of some function spaces

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6338524)