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first countable spaces

hksuraj

New member
Feb 12, 2017
1
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
Jun 20, 2014
1,890
Welcome, hksuraj ! (Wave)

What have you tried? Please show your thoughts on this problem, even if you only have the definitions.
 

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,890
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.