Continuous image of compact set is compact

Continuous image of compact set is compact: Let $f:X\to Y$ be a continuous map between topological spaces . If $K\subseteq X$ is compact , then $f(K)$ is compact in $Y$, where $f(K)$ is the image of $K$ under the function $f$.

This is one of the most important invariance properties in topology; it underlies results like attainment of maxima and minima on compact domains .