Open and Closed Maps
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets. (The concept of a closed map should not be confused with that of a closed operator.) Neither open nor closed maps are required to be continuous. Although their definitions seem natural, open and clo...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In topology, an open map is a function between two topological spaces which maps open sets to open sets. That is, a function f : X Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function which maps closed sets to closed sets. (The concept of a closed map should not be confused with that of a closed operator.) Neither open nor closed maps are required to be continuous. Although their definitions seem natural, open and closed maps are much less important than continuous maps. Recall that a function f : X Y is continuous if the preimage of every open set of Y is open in X. (Equivalently, if the preimage of every closed set of Y is closed in X).
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