Observations on spaces with property $(DC(\omega_1))$

Abstract

A topological space $X$ has property $(DC(\omega_1))$ if it has a dense subspace every uncountable subset of which has a limit point in $X$. In this paper, we make some observations on spaces with property $(DC(\omega_1))$. In particular, we prove that the cardinality of a space $X$ with property $(DC(\omega_1))$ does not exceed $\mathfrak c$ if $X$ satisfies one of the following conditions: (1) $X$ is normal and has a rank $2$-diagonal; (2) $X$ is perfect and has a rank $2$-diagonal; (3) $X$ has a rank $3$-diagonal; (4) $X$ is perfect and has countable tightness. We also prove that if $X$ is a regular space with a $G_\delta$-diagonal and property $(DC(\omega_1))$ then the cardinality of $X$ is at most $2^\mathfrak c$.