Exercise 3.3.13

Question

Let's call a set \emph{clompact} if it has the property that every \emph{closed} cover (i.e., a cover consisting of closed sets) admits a finite subcover. Describe all of the clompact subsets of $\mathbf{R}$.

Ulisse Mini · Accepted Answer

$K$ is clompact if and only if $K$ is finite, since the closed cover $\{[x,x] : x \in K\} = K$ having a finite subcover implies $\{[x,x] : x \in K\}$ is finite (since it is the only subcover that works) therefore $K$ is finite. If $K$ is finite then it obviously permits a finite subcover.

Exercise 3.3.13

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Exercise 3.3.13

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