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Given a compact Hausdorff space $X$, if $a \in X$ is a $G_\delta$ singleton (i.e., $\{ a \}$ is a $G_\delta$ set), then there is a countable local base at $a$.

The point $a$ can be written as countable intersection of open sets let's call $U_{n}$.

How can I use the compactness? What is the next step for proof? Could you give me hint please?

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$a$ is a point. Not a set. Did you mean $\{a\}$? Usually one assumes $a\neq\{a\}$. –  Asaf Karagila Nov 19 '12 at 8:50
    
You are exactly right, I mean that single set. thanks. –  ege Nov 19 '12 at 9:21

1 Answer 1

up vote 4 down vote accepted

$\newcommand{\cl}{\operatorname{cl}}$Suppose that $X$ is a compact Hausdorff space, $a\in X$, and $\{a\}=\bigcap_{n\in\Bbb N}U_n$, where each $U_n$ is open in $X$. Let $V_0=U_0$. A compact Hausdorff space is regular, so there is an open $V_1$ such that $$a\in V_1\subseteq\cl V_1\subseteq V_0\cap U_1\;.$$ Continue in this fashion: given $V_n$, let $V_{n+1}$ be an open set such that $$a\in V_{n+1}\subseteq\cl V_{n+1}\subseteq V_n\cap U_{n+1}\;.$$ Then $\{a\}=\bigcap_{n\in\Bbb N}V_n=\bigcap_{n\in\Bbb N}\cl V_n$, and $$V_0\supseteq\cl V_1\subseteq V_1\supseteq\cl V_2\supseteq V_2\supseteq\ldots~\;.$$

For each $n\in\Bbb N$ let $W_n=X\setminus\cl V_n$, and let $\mathscr{W}=\{W_n:n\in\Bbb N\}$; then $\mathscr{W}$ is an open cover of $X\setminus\{a\}$, and $W_0\subseteq W_1\subseteq W_2\subseteq\ldots~$.

Now suppose that $U$ is an open set containing $a$, and let $F=X\setminus U$; $a\notin F$, so $\mathscr{W}=\{W_n:n\in\Bbb N\}$ is an open cover of $F$. Moreover, $F$ is closed and therefore compact, so a finite subfamily of $\mathscr{W}$ covers $F$. Use this to show that there is an $n\in\Bbb N$ such that $a\in V_n\subseteq U$ and conclude that $\{V_n:n\in\Bbb N\}$ is a base at $a$.

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what is the $V_0$? –  ege Nov 19 '12 at 9:20
    
@ege: Sorry: I did a lot of editing while I was writing, and that sentence got lost. It’s back now. –  Brian M. Scott Nov 19 '12 at 9:22
    
$V_1 \subset cl V_1$. –  ege Nov 19 '12 at 9:32
    
Before the sentence with began For each....,$V_0$ contain $cl V_1$ contain $V_1$..... –  ege Nov 19 '12 at 9:36
    
@ege: Yes, both of those are correct. –  Brian M. Scott Nov 19 '12 at 9:37

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