The Unapologetic Mathematician

Mathematics for the interested outsider

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 Posted by | Point-Set Topology, Topology | 6 Comments