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- W4363672029 abstract "Assume that $mathcal{P}$ is a topological property of a space $X$, then we say that $X$ is {it dense-$mathcal{P}$} if each dense subset of $X$ has the property $mathcal{P}$. In this paper, we mainly discuss dense subsets of a space $X$, and we prove that: (1) if $X$ is Tychonoff space, then $X$ is dense-pseudocompact iff the range of each continuous real-valued function $f$ on $X$ is finite, iff $X$ is finite, iff $X$ is hereditarily pseudocompact; (2) $X$ is dense-connected iff $overline{U}=X$ for any non-empty open subset $U$ of $X$; (3) $X$ is dense-ultraconnected iff for point $xin X$, we have $overline{{x}}=X$ or ${x}cup (Xsetminusoverline{{x}})$ is the unique open neighborhood of $x$ in ${x}cup (Xsetminusoverline{{x}})$, iff for any two points $x$ and $y$ in $X$, we have $xin overline{{y}}$ or $yin overline{{x}}$. Moreover, we give a characterization of a topological group (resp., paratopological group, quasi-topological group) $G$ such that $G$ is dense-connected." @default.
- W4363672029 created "2023-04-11" @default.
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- W4363672029 date "2023-04-06" @default.
- W4363672029 modified "2023-09-27" @default.
- W4363672029 title "The characterizations of dense-pseudocompact and dense-connected spaces" @default.
- W4363672029 doi "https://doi.org/10.48550/arxiv.2304.03457" @default.
- W4363672029 hasPublicationYear "2023" @default.
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