I'm a beginner on topology. I was reading Munkres' Analysis on Manifolds when I came across the following theorem (or a definition of compactness).
A subspace $X$ of $R^n$ is compact if and only if for every collection of sets open in $X$ whose union is $X$, there is a finite subcollection whose union equals $X$.
Is it generally true that for every compact subspace $X$ of $R^n$ there exists a finite collection of sets open in $X$ whose union equals $X$?
There are many finite unions of open sets (in $X$) that contain $X$, but it is trivial that we can really find one among those unions that is equal to $X$?
Any help will be appreciated.
Take any finite cover of $X$ in $\mathbb{R}^n$ and take the intersections of those sets with $X$. Then those will be open in $X$ (in the subspace topology - this is probably how he means you to interpret this) and their union will be $X$.