m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS

سال انتشار: 1400
نوع سند: مقاله ژورنالی
زبان: انگلیسی
مشاهده: 298

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شناسه ملی سند علمی:

JR_JAS-9-1_008

تاریخ نمایه سازی: 13 اردیبهشت 1400

چکیده مقاله:

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 $M_m(X,\mathscr{A})$.

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نویسندگان

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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