m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS
Publish place: Journal of Algebraic Systems، Vol: 9، Issue: 1
Publish Year: 1400
نوع سند: مقاله ژورنالی
زبان: English
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JR_JAS-9-1_008
تاریخ نمایه سازی: 13 اردیبهشت 1400
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
Authors
H. Yousefpour
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
A. A. Estaji
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
A. Mahmoudi Darghadam
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Gh. Sadeghi
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
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