Abstract:We prove that if $X$ is a first countable space with property $(DC(\omega_1))$ and with a $G_\delta$-diagonal then the cardinality of $X$ is at most $\mathfrak c$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $\mathfrak c$.
Keywords: $G_\delta$-diagonal; property $(DC(\omega_1))$; cardinal; DCCC
DOI: DOI 10.14712/1213-7243.2015.190
AMS Subject Classification: 54D20 54E35