Shahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS83106205610.22044/jas.2020.9557.1470ENH. YousefpourFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.A. A. EstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.0000-0001-8993-5109A. Mahmoudi DarghadamFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.Gh. SadeghiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.Journal Article20200410In this article we consider the $m$-topology on <br /> \linebreak <br /> $M(X,\mathscr{A})$, the ring of all real measurable functions on a measurable space<br /> $(X, \mathscr{A})$, and we denote it by $M_m(X,\mathscr{A})$.<br /> 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. <br /> 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 $M_m(X,\mathscr{A})$.https://jas.shahroodut.ac.ir/article_2056_ea0b901cccd560776fdc7441db04840b.pdf