Compactness is continuous image-closed

Statement
Under a continuous map, the image of a compact space is a compact space. (Note that if the map is not surjective, we mean by the image the subset of the space on the right to which the map goes, endowed with the subspace topology).

Applications

 * For functions to the real line: Recall that a subset of the real line is compact iff it is closed and bounded, which implies that it has a unique maximum and minimum. Thus, any continuous real-valued function on a compact space attains an absolute maximum and an absolute minimum. This result is sometimes called the extreme-value theorem. Topological spaces that satisfy the conclusion of the theorem are termed pseudocompact spaces -- there do exist non-compact pseudocompact spaces.
 * For functions from closed intervals: As a further corollary of the above, the image, under a continuous function, of a closed and bounded interval in the real line attains an absolute maximum and an absolute minimum. Combining this with the fact that connectedness is continuous image-closed, which gives rise to the intermediate-value theorem, we obtain that the image of a closed and bounded interval under a continuous map is again a closed and bounded interval.
 * Compact to Hausdorff implies closed: Any map from a compact space to a Hausdorff space is a closed map.