There is a well-known result in topology,
Any continuous bijection from a compact topological space to a Hausdorff space is a homeomorphism.
I was wondering whethet the following (slightly weaker) statement holds:
Let $K$ be a compact topological space and $X$ a topological space. Then $f(K)$ is compact in $X$ for any continuous map $f\colon K\to X$.