@article {
author = {Yousefpour, H. and Estaji, A. A. and Mahmoudi Darghadam, A. and Sadeghi, Gh.},
title = {m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS},
journal = {Journal of Algebraic Systems},
volume = {9},
number = {1},
pages = {83-106},
year = {2021},
publisher = {Shahrood University of Technology},
issn = {2345-5128},
eissn = {2345-511X},
doi = {10.22044/jas.2020.9557.1470},
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 = {m-topology,measurable space,pseudocompact measurable space,connected space,first countable topological space},
url = {https://jas.shahroodut.ac.ir/article_2056.html},
eprint = {https://jas.shahroodut.ac.ir/article_2056_ea0b901cccd560776fdc7441db04840b.pdf}
}