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Let $X$ be a topological space. True or false?

$X$ is metrizable if and only if it contains a sequence of metrizable spaces $\{X_n\}$ with $X=\bigcup X_n$!

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    $\begingroup$ Please explain, what does it mean a sequence of spaces? $\endgroup$
    – Evgeny Kuznetsov
    Commented Apr 26, 2018 at 8:22
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    $\begingroup$ The line with double origin is a union of two metrizable spaces, but is not metrizable itself, as it is not Hausdorff. $\endgroup$
    – user1688
    Commented Apr 26, 2018 at 8:26
  • $\begingroup$ @ Evgeny Kuznetsov I mean $X_n$ is just contained in $X$ for every $n$. $\endgroup$
    – ABB
    Commented Apr 26, 2018 at 8:30
  • $\begingroup$ @ Corbennick Nice example. $\endgroup$
    – ABB
    Commented Apr 26, 2018 at 8:35
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    $\begingroup$ Isn't every countable $\text{T}_1$-space the union of a sequence of metrizable spaces? $\endgroup$
    – bof
    Commented Apr 26, 2018 at 9:17

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