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 -