# first countable spaces

#### hksuraj

##### New member
Show that in a first countable hausdorff space every one point set is a $G_\delta$ set. How to prove this?

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#### Euge

##### MHB Global Moderator
Staff member
Welcome, hksuraj ! What have you tried? Please show your thoughts on this problem, even if you only have the definitions.

#### Euge

##### MHB Global Moderator
Staff member
For the sake of completion, I'll post a complete answer. Let $X$ be a first countable Hausdorff space. Fix $x\in X$, and let $\{B_n\}_{n\in \Bbb N}$ be a countable basis at $x$. For every $y\neq x$, there are disjoint open sets $U_y, V_y$ containing $x$ and $y$, respectively. Each $U_y$ contains $B_{n(y)}$ for some index $n(y)$. Let $G = \bigcap B_{n(y)}$. Being the countable intersection of open sets, $G$ is a $G_\delta$ set. Since $x\in G$ and

$$X\setminus G = \bigcup_{y\neq x} X\setminus B_{n(y)} \supset \bigcup_{y\neq x} V_y \supset X\setminus\{x\}$$

then $\{x\} = G$. Hence, $\{x\}$ is a $G_\delta$ set.