On the definition of compactness

Question

Rudin, in Principles of Mathematical Analysis, defines compactness: A set in a metric space is compact if and only if for any open cover $\{G_\alpha\}$ of $E$ there exist a finite subcover $G_{\alpha_1},...,G_{\alpha_k}$ such that:

$E \subseteq G_{\alpha_1} \cup \cdots \cup G_{\alpha_k}.$

My question is about changing the word any for some, i think that would be valid, any suggestion?

Answer 1

Consider the open cover $\{X\}$. This is a finite open cover of every subset of $X$. So your modified notion would apply to every set.

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That said, there is a version of this notion which is nontrivial: what if we bound the sizes of the open sets involved? (Note that this is a fundamentally metric, as opposed to topological, idea.) Specifically, consider the following property:

For each $\epsilon>0$ there is a finite cover of $X$ consisting only of open sets with diameter $<\epsilon$.

Here the diameter of an open set $U$ is the supremum of the distances between elements of that set:$$diam(U)=\sup\{d(a,b): a,b\in U\}.$$

This is no longer trivial - e.g. $\mathbb{R}$ with the usual metric does not have this property. But this is still very different from compactness.

Answer 2

If you do that, then every subset of $E$ of $X$ has that property. Just take the open cover $\{X\}$ of $E$. It has a finite subcover (itself).