m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS | ||
| Journal of Algebraic Systems | ||
| دوره 9، شماره 1، آذر 2021، صفحه 83-106 اصل مقاله (234.06 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2020.9557.1470 | ||
| نویسندگان | ||
| H. Yousefpour؛ A. A. Estaji* ؛ A. Mahmoudi Darghadam؛ Gh. Sadeghi | ||
| Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. | ||
| چکیده | ||
| 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 $M_m(X,\mathscr{A})$. | ||
| کلیدواژهها | ||
| m-topology؛ measurable space؛ pseudocompact measurable space؛ connected space؛ first countable topological space | ||
| مراجع | ||
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