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show that in a first countable hausdorff space every one point set is g delta set?I want the proof of this above question

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    That's not how this site works – 2017-02-11
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    See the post [How to ask a good question](http://meta.math.stackexchange.com/q/9959), especially the section on [adding context](http://meta.math.stackexchange.com/a/9960). – 2017-02-11

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Suppose $X = \{x_n: n \in \mathbb{N}\}$ to make it explicitly countable. As $X$ is $T_1$ in particular, finite sets are closed, so all sets $U_n = X\setminus\{x_n\}$ are open. Now $$\{x_m\} =\cap \{U_n: n \neq m\}$$ for every $m$, making all singletons a $G_\delta$.

Note that Hausdorff is slightly overkill, only $T_1$ is used.