Document Type : Original Manuscript

Authors

Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

Abstract

In this article we consider the $m$-topology on
\linebreak
$M(X,\mathscr{A})$, the ring of all real measurable functions on a measurable space
$(X, \mathscr{A})$, and we denote it by $M_m(X,\mathscr{A})$.
We show that $M_m(X,\mathscr{A})$ is a Hausdorff regular topological ring, moreover we prove that if $(X, \mathscr{A})$ is a $T$-measurable space and $X$ is a finite set with $|X|=n$, then $M_m(X,\mathscr{A})‎\cong‎ \mathbb R^n$ as topological rings.
Also, we show that $M_m(X,\mathscr{A})$ is never a pseudocompact space and it is also never a countably compact space. We prove that $(X,\mathscr{A})$ is a pseudocompact measurable space, if and only if $ {M}_{m}(X,\mathscr{A})= {M}_{u}(X,\mathscr{A})$, if and only if $ M_m(X,\mathscr{A}) $ is a first countable topological space, if and only if $M_m(X,\mathscr{A})$ is a connected space, if and only if $M_m(X,\mathscr{A})$ is a locally connected space, if and only if $M^*(X,\mathscr{A})$ is a connected subset of $M_m(X,\mathscr{A})$.

Keywords

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