The characterizations of dense-pseudocompact and dense-connected spaces

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 $x\in X$, we have $\overline{\{x\}}=X$ or $\{x\}\cup (X\setminus\overline{\{x\}})$ is the unique open neighborhood of $x$ in $\{x\}\cup (X\setminus\overline{\{x\}})$, iff 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.