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Closed Map

A map f between topological spaces that maps closed sets to closed sets. If f is bijective, then

f is closed <==>f is open <==>f^(-1) is continuous,

where f^(-1) denotes the inverse map. In particular, a homeomorphism can be characterized as a continuous bijection which is open (or, equivalently, closed).

This entry contributed by Margherita Barile

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Cite this as:

Weisstein, Eric W., with contributions by Margherita Barile. "Closed Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedMap.html

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