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# Article

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Keywords:
$G_\delta$-diagonal; property $(DC(\omega_1))$; cardinal; DCCC
Summary:
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 $\mathfrak c$.
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