Useful Properties of Compact Spaces and Continuous Functions
Theorem
For $f : X \to Y$, suppose $X$ is compact and $f$ is continuous.
- [1]: If $f$ is surjective, then $Y$ is compact. Even if $f$ is not surjective, $f(X)$ is compact.
- [2]: If $Y$ is Hausdorff, then $f$ is a closed function. For a closed set $C \subset X$, $f(C) \subset Y$ is a closed set.
- [3]: If $f$ is a bijection and $Y$ is Hausdorff, then $f$ is a homeomorphism.
- [4]: If $X$ is a metric space, then $f$ is uniformly continuous.
Explanation
You might think these are just trivial properties, but they have many uses, such as in proving the extreme value theorem.
[1]
This means that compactness is a property that is still preserved when a continuous function is applied.
[2]
Although it talks about closedness, since it is really about Hausdorff spaces, it is used all over the place. Every metric space is a $T_{2}$ space, so it is safe to assume it works in most cases.
[3]
There are many conditions, but the key point is that the definition of a homeomorphism implies that the inverse function is also continuous. It would be very useful when it is easier to grasp the topological properties of the domain and codomain than to directly check the continuity of the inverse function.
[4]
Uniform continuity is originally a concept discussed only in metric spaces, but when $X$ becomes a compact space, it guarantees that $f$ has not only continuity but also uniform continuity, which is useful. Compactness and continuity start from very different concepts, yet they are intertwined in so many ways like this that they cannot be separated.