# The Unapologetic Mathematician

## The Image of a Compact Space

One of the nice things about connectedness is that it’s preserved under continuous maps. It turns out that compactness is the same way — the image of a compact space $X$ under a continuous map $f:X\rightarrow Y$ is compact.

Let’s take an open cover $\{U_i\}$ of the image $f(X)$. Since $f$ is continuous, we can take the preimage of each of these open sets $\{f^{-1}(U_i)\}$ to get a bunch of open sets in $X$. Clearly every point of $X$ is the preimage of some point of $f(X)$, so the $f^{-1}(U_i)$ form an open cover of $X$. Then we can take a finite subcover by compactness of $X$, picking out some finite collection of indices. Then looking back at the $U_i$ corresponding to these indices (instead of their preimages) we get a finite subcover of $f(X)$. Thus any open cover of the image has a finite subcover, and the image is compact.

January 16, 2008