Compacttocompactness

Compacttocompactness, or compact-to-compactness, is a proposed property of a function f: X -> Y between topological spaces. A map is called compact-to-compact if the image f(K) of every compact subset K of X is a compact subset of Y. The notion is designed to isolate how a function handles compact sets, independently of other topological factors.

In practice, compact-to-compactness is automatic for any continuous map, because the image of a compact set under a continuous function is always compact. Therefore the concept is mainly notable when considering discontinuous or specially defined maps, where the image of a compact set need not be compact. For example, let X be the subset {0} ∪ {1/n : n ∈ N} of the real line with the subspace topology, and define f by f(0) = 0 and f(1/n) = n. The set X is compact, but the image f(X) = {0} ∪ {n : n ∈ N} is not compact in the real line, so f is not compact-to-compact. This illustrates that the property is not guaranteed without some form of continuity.

Related properties include composition and restriction: the composition of two compact-to-compact maps is compact-to-compact, and restricting a compact-to-compact map to a subspace remains compact-to-compact. The concept has limited standardization and is generally discussed informally as a way to study how maps preserve compactness on their domain.

Notes: compact-to-compactness is distinct from proper maps, which concern preimages of compact sets rather than images. It is not a widely standardized term in mainstream topology, but it serves as a useful descriptive notion in discussions of image-preserving behavior.