TY - JOUR ID - 2056 TI - m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS JO - Journal of Algebraic Systems JA - JAS LA - en SN - 2345-5128 AU - Yousefpour, H. AU - Estaji, A. A. AU - Mahmoudi Darghadam, A. AU - Sadeghi, Gh. AD - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Y1 - 2021 PY - 2021 VL - 9 IS - 1 SP - 83 EP - 106 KW - m-topology KW - measurable space KW - pseudocompact measurable space KW - connected space KW - first countable topological space DO - 10.22044/jas.2020.9557.1470 N2 - 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 $M_m(X,\mathscr{A})$. UR - https://jas.shahroodut.ac.ir/article_2056.html L1 - https://jas.shahroodut.ac.ir/article_2056_ea0b901cccd560776fdc7441db04840b.pdf ER -