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The characterizations of dense-pseudocompact and dense-connected spaces. (English) Zbl 07695411

Summary: Assume that \(\mathcal{P}\) is a topological property of a space \(X\), then we say that \(X\) is 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 if and only if the range of each continuous real-valued function \(f\) on \(X\) is finite, if and only if \(X\) is finite, if and only if \(X\) is hereditarily pseudocompact;
(2) \(X\) is dense-connected if and only if \(\overline{U} = X\) for any non-empty open subset \(U\) of \(X\);
(3) \(X\) is dense-ultraconnected if and only if for point \(x \in X\), we have \(\overline{\{ x \}} = X\) or \(\{x \} \cup(X \setminus \overline{\{ x \}})\) is the unique open neighborhood of \(x\) in \(\{x \} \cup(X \setminus \overline{\{ x \}})\), if and only if for any two points \(x\) and \(y\) in \(X\), we have \(x \in \overline{\{ y \}}\) or \(y \in \overline{\{ x \}} \).
Moreover, we give a characterization of a topological group (resp., paratopological group, quasi-topological group) \(G\) such that \(G\) is dense-connected.

MSC:

22A05 Structure of general topological groups
54B05 Subspaces in general topology
54C30 Real-valued functions in general topology
54D05 Connected and locally connected spaces (general aspects)
54H11 Topological groups (topological aspects)
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