# Compactness is continuous image-closed

From Topospaces

This article gives the statement, and possibly proof, of a topological space property (i.e., compact space) satisfying a topological space metaproperty (i.e., continuous image-closed property of topological spaces)

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## 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).

## Related facts

### 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.

## Proof

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