%0 Journal Article
%T m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS
%J Journal of Algebraic Systems
%I Shahrood University of Technology
%Z 2345-5128
%A Yousefpour, H.
%A Estaji, A. A.
%A Mahmoudi Darghadam, A.
%A Sadeghi, Gh.
%D 2021
%\ 09/01/2021
%V 9
%N 1
%P 83-106
%! m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS
%K m-topology
%K measurable space
%K pseudocompact measurable space
%K connected space
%K first countable topological space
%R 10.22044/jas.2020.9557.1470
%X 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 $M_m(X,\mathscr{A})$.
%U https://jas.shahroodut.ac.ir/article_2056_ea0b901cccd560776fdc7441db04840b.pdf