A compact space whose proper compact subspaces are finite A recent deleted question asked:

Are there infinite compact spaces whose only compact subsets are finite or itself?


The answer is no.  Suppose $X$ is such a space.  Since closed subsets of compact spaces are themselves compact, every closed proper subset of $X$ is compact and therefore finite.  
Now let $S$ be an arbitrary infinite subset of $X$ and let $\mathcal C$ be an open cover of $S$. Suppose $G$ is any element of $\mathcal C$.  Then  $S\setminus G$ is finite because it is the intersection of $S$ and the closed (and therefore finite) set $X\setminus G$.  The finite set $S\setminus G$ is easily covered by a finite subcover $\mathcal C'\subset \mathcal C$, so  $\mathcal C'\cup\{G\}$ is a finite subcover of $S$.  This shows that the infinite set $S$ is compact, a contradiction.
