the continuous image of a compact space is compact
Consider a continuous and surjective function and a compact set. We will prove that is also a compact set.
Let be an open covering of . By the continuity of the pre-image by of any open subset (http://planetmath.org/OpenSubset) of will also be an open subset of .So we have an open covering of where .
To see this remember, since the continuity of implies that each is open, all we need to prove is that .Consider , we know that since is a covering of that there exists such that but then by construction and is indeed an open covering of .
Since is compact we can consider a finite set of indices such that is a finite open covering of , but then will be a finite open covering of and it will thus be a compact set.
To see that is a covering of consider . By the surjectivity of there must exist (at least) one such that and since is a finite covering of , there exists such that .But then since , we must have that and is indeed a finite open covering of .