Wrong conclusion on metric space $(X, d)$ where every closed, bounded set is compact

Question

Suppose $(X,d)$ is a metric space where every closed, bounded set is compact. I was trying to prove that every open cover of X admits a countable subcover, but came to the conclusion:

If:

X=$\bigcup_{\alpha \in \Delta} U_{\alpha}$

=$\bigcup_{\alpha \in \Delta} \bigcup_{n \geq 1}$ $[ U_{\alpha}\cap B[x,n] ]$=

=$\bigcup_{n \geq 1}$ ($\bigcup_{\alpha \in \Delta}$ $[ U_{\alpha}\cap B[x,n] ])$=$(*)$

$\bigcup_{\alpha \in \Delta}$ $( U_{\alpha}\cap B[x,n] )$ is an open cover of B[x,n], which is compact by hypothesis, therefore has finite subcover.

So:

$(*)$= $\bigcup_{n \geq 1}$ $\bigcup_{j \in {1,...,m}} $ $[ U_{j}\cap B[x,n] ]$=

=$\bigcup_{j \in {1,...,m}}$ $\bigcup_{n \geq 1}$ $[ U_{j}\cap B[x,n] ]$=

=$\bigcup_{j \in {1,...,m}}$ $U_{j}$

So we concluded X is compact, but this is not true for the set of reals, for example, so something must be wrong... If someone could help me finding the error, I'd really appreciate it.

Thank you in advance.

Answer 1

First of all, there is a (correctable) problem in the claim that

$\bigcup_{\alpha \in \Delta} U_\alpha = \bigcup_{\alpha \in \Delta} \bigcup_{n \geq 1} [U_{\alpha} \cap B[x,n]]$

This doesn't make sense to write because you haven't said what $x$ is. Can you rewrite the start of your proof to fix this?

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Now here is the real problem. You say

$\bigcup_{\alpha \in \Delta}$ $( U_{\alpha}\cap B[x,n] )$ is an open cover of B[x,n], which is compact by hypothesis, therefore has finite subcover.

True! But more precisely, you should say that for each $n$, $\bigcup_{\alpha \in \Delta}$ $( U_{\alpha}\cap B[x,n] )$ is an open cover of $B[x,n]$. Then for each $n$ we can pick a finite subcover, but we will get (potentially) a different subcover for each $n$!

In contrast, you wrote

...
$\bigcup_{n \geq 1}$ $\bigcup_{j \in {1,...,m}} $ $[ U_{j}\cap B[x,n] ]$

Meaning that you are using the same finite subcover $\{U_1, \dots, U_m\}$ for each $n$! This is not valid.

Also, the symbols $U_1, \dots, U_m$ don't really make sense, since we don't know that $1, \dots, m \in \Delta$.