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Concept
Open and closed maps
Summary
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function is open if for any open set in the is open in
Likewise, a closed map is a function that maps closed sets to closed sets.
A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.
Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces.
Although their definitions seem more natural, open and closed maps are much less important than continuous maps.
Recall that, by definition, a function is continuous if the of every open set of is open in (Equivalently, if the preimage of every closed set of is closed in ).
Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.
If is a subset of a topological space then let and (resp. ) denote the closure (resp. interior) of in that space.
Let be a function between topological spaces. If is any set then is called the image of under
There are two different competing, but closely related, definitions of "" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets."
The following terminology is sometimes used to distinguish between the two definitions.
A map is called a
"" if whenever is an open subset of the domain then is an open subset of 's codomain
"" if whenever is an open subset of the domain then is an open subset of 's where as usual, this set is endowed with the subspace topology induced on it by 's codomain
Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.
Warning: Many authors define "open map" to mean " open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean " open map".
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