A number of properties enjoyed by two specially constructed topologies on $C(X)$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917226313940992 |
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| author | Dey, Soumajit Acharyya, Sudip Kumar Mandal, Dhananjoy |
| author_facet | Dey, Soumajit Acharyya, Sudip Kumar Mandal, Dhananjoy |
| contents | If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$, the $m^I$-topology and $u^I$-topology on $C(X)$, generalizing the well-known $m$-topology and $u$-topology in $C(X)$ respectively are already there in the literature. It is proved amongst others that the $m^I$-topology is first countable if and only if the $u^I$-topology= $m^I$-topology on $C(X)$ if and only if $X$ is $I$-$pseudocompact$. A special case of this result on choosing $I=C(X)$ reads: the $u$-topology and $m$-topology on $C(X)$ coincide if and only if $X$ is pseudocompact. It is established that the $m^I$-topology on $C(X)$ is second countable if and only if it is $\aleph_0$-$bounded$ if and only if $X$ is compact, metrizable and $I=C(X)$. Furthermore it is realized that the $m^I$ topology on $C(X)$ is hemicompact if and only if it is $σ$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14651 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A number of properties enjoyed by two specially constructed topologies on $C(X)$ Dey, Soumajit Acharyya, Sudip Kumar Mandal, Dhananjoy General Topology Primary 54C40, Secondary 46E30 If $I$ is an ideal in the ring $C(X)$ of all real valued continuous functions defined over a Tychonoff space $X$, then $X$ is called $I$-$pseudocompact$ if the set $X\setminus \bigcap Z[I]$ is a bounded subset of $X$. Corresponding to $I$, the $m^I$-topology and $u^I$-topology on $C(X)$, generalizing the well-known $m$-topology and $u$-topology in $C(X)$ respectively are already there in the literature. It is proved amongst others that the $m^I$-topology is first countable if and only if the $u^I$-topology= $m^I$-topology on $C(X)$ if and only if $X$ is $I$-$pseudocompact$. A special case of this result on choosing $I=C(X)$ reads: the $u$-topology and $m$-topology on $C(X)$ coincide if and only if $X$ is pseudocompact. It is established that the $m^I$-topology on $C(X)$ is second countable if and only if it is $\aleph_0$-$bounded$ if and only if $X$ is compact, metrizable and $I=C(X)$. Furthermore it is realized that the $m^I$ topology on $C(X)$ is hemicompact if and only if it is $σ$-compact if and only if this topology is $H$-$bounded$ if and only if $X$ is finite and $I=C(X)$. |
| title | A number of properties enjoyed by two specially constructed topologies on $C(X)$ |
| topic | General Topology Primary 54C40, Secondary 46E30 |
| url | https://arxiv.org/abs/2508.14651 |