Geodesic
Connectedness of some rings of quotients of $C(X)$ with the $m$-topology
Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 1, pp. 63-76 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this article we define the $m$-topology on some rings of quotients of $C(X)$. Using this, we equip the classical ring of quotients $q(X)$ of $C(X)$ with the $m$-topology and we show that $C(X)$ with the $r$-topology is in fact a subspace of $q(X)$ with the $m$-topology. Characterization of the components of rings of quotients of $C(X)$ is given and using this, it turns out that $q(X)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C(X)$ with $r$-topology is connected. We also observe that the maximal ring of quotients $Q(X)$ of $C(X)$ with the $m$-topology is connected if and only if $X$ is finite. Finally for each point $x$, we introduce a natural ring of quotients of $C(X)/O_x$ which is connected with the $m$-topology.
In this article we define the $m$-topology on some rings of quotients of $C(X)$. Using this, we equip the classical ring of quotients $q(X)$ of $C(X)$ with the $m$-topology and we show that $C(X)$ with the $r$-topology is in fact a subspace of $q(X)$ with the $m$-topology. Characterization of the components of rings of quotients of $C(X)$ is given and using this, it turns out that $q(X)$ with the $m$-topology is connected if and only if $X$ is a pseudocompact almost $P$-space, if and only if $C(X)$ with $r$-topology is connected. We also observe that the maximal ring of quotients $Q(X)$ of $C(X)$ with the $m$-topology is connected if and only if $X$ is finite. Finally for each point $x$, we introduce a natural ring of quotients of $C(X)/O_x$ which is connected with the $m$-topology.
DOI : 10.14712/1213-7243.015.106
Classification : 54C35, 54C40
Keywords: $r$-topology; $m$-topology; almost $P$-space; pseudocompact space; component; classical ring of quotients of $C(X)$ 
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     title = {Connectedness of some rings of quotients of $C(X)$ with the $m$-topology},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {63--76},
     year = {2015},
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     language = {en},
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Azarpanah, F.; Paimann, M.; Salehi, A. R. Connectedness of some rings of quotients of $C(X)$ with the $m$-topology. Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 1, pp. 63-76. doi: 10.14712/1213-7243.015.106

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