TITLE: Finding all compact subsets of a given set
QUESTION [0 upvotes]: Let $K=\{0\} \cup \{ \frac{1}{n} : n \in \mathbb{Z} ^{+} \}$. I know that $K$ itself is compact, because for every open cover there is a finite subcover which must include the open set that contains $0$. $0$ being an element of $K$ is crucial. But I am stuck for finding a criterion to describe all compact subsets of $K.$

REPLY [0 votes]: If $C \subseteq K$ is a subset then $C$ is compact iff $C$ is finite or $0 \in C$.
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