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Prove that a compact metric space has a countable base. $$$$Let $X$ be a compact metric space. Suppose for every point $x$ in $X$ we take the open n-ball $B(x, 1)$ with $x$ as the centre and radius $1$ units. Then we have $$X \subset \bigcup_{x \in X}B(x, 1)$$. Now as $X$ is compact, so a finite number of these open n-balls will cover $X$ and hence let $$X \subset \bigcup_{k=1}^{m}B(x_k, 1)$$. Now in each of these finite number of open n-balls we chose our collection of open sets as follows: In each of these open n-balls chose all the points with rational coordinates and their all the neighbourhoods with rational radii. Then it is easy to see that this collection of open sets is a countable collection. Now consider any open set $G$ in $X$ such that $x \in G$ then, $$x \in B(x_j, 1)$$ for some $1 \leq j \leq m$. Now as $x \in G$ so there is an open n-ball of $x$ such that $$B(x, r) \subset G$$. Now chose a point $a$ with rational coordinates such that it belongs to any one of these open n-balls and $$d(a, x) < \frac{r}{2}$$. Now chose a neighbourhood of $a$ with rational radii less than $\frac{r}{2}$ which contains $x$. Then this open set say $C$ belongs to our chosen collection and also belongs to $B(x, r)$ and as $$B(x, r) \subset G$$, so we have an open set from our chosen collection such that $$x \in C \subset G$$ and hence our collection is a countable base for $X$. $$$$Is My Proof Correct??

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  • $\begingroup$ "In each of these open n-balls chose all the points with rational coordinates and their all the neighbourhoods with rational radii." What does rational coordinates mean in a metric space? $\endgroup$
    – N. S.
    Sep 25 '20 at 17:16
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The part "in each of these open n-balls chose all the points with rational coordinates" doesn't make sense in abstract metric spaces.

Hint Instead of covering $X$ with balls of radius 1, try to cover it with balls of radius $\frac{1}{n}$ for all $n$.

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  • $\begingroup$ But why points with rational coordinates doesn't make sense in metric spaces? $\endgroup$
    – user763338
    Sep 25 '20 at 17:26
  • $\begingroup$ @user763338 Your metric space is a SET, with a metric. Your set could be the set of all dogs in the world with the distance being the distance between these dogs at some moment in time. Now, which are the dogs with "rational coordinates"? $\endgroup$
    – N. S.
    Sep 25 '20 at 17:37

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