## 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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### References:

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