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Please help me to prove the following statement:

In a first countable $T_1$ space, every point is a $G_\delta$ set.

A space is first countable if it has a countable base. Also in a $T_1$ space every singleton is closed. So how can I use these to prove the above statement?

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    You might also take a look at [$G_\delta$ singletons in compact Hausdorff and first countability](http://math.stackexchange.com/questions/240472/g-delta-singletons-in-compact-hausdorff-and-first-countability) for sort of the converse statement.2012-11-22

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