noncompact

Noncompact is an adjective used in topology to describe a space that is not compact. Compactness means that every open cover has a finite subcover; equivalently, any collection of open sets that covers the space contains a finite subcollection that also covers it. Compactness is a topological property, preserved under homeomorphisms.

In Euclidean spaces, the Heine-Borel theorem provides a practical criterion: a subset of R^n is compact if and only if it is closed and bounded. Therefore noncompactness can arise from unboundedness or from not being closed. Examples include the real line R, the open interval (0,1), and the plane R^2. By contrast, closed and bounded intervals such as [0,1] are compact.

Many spaces are noncompact yet retain useful structure. For instance, R is locally compact and sigma-compact. A discrete infinite space is noncompact (there is an open cover by singletons with no finite subcover), whereas any finite discrete space is compact.

Noncompactness is preserved by homeomorphisms: if a space X is noncompact and X is homeomorphic to a space Y, then Y is noncompact as well. In practice, mathematicians study noncompact spaces through compactifications, such as the one-point compactification, which embeds a noncompact space as a dense open subset of a compact space.

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