Open and closed maps
A function that sends open (resp. closed) subsets to open (resp. closed) subsets / From Wikipedia, the free encyclopedia
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.[1][2][3]
That is, a function is open if for any open set
in
the image
is open in
Likewise, a closed map is a function that maps closed sets to closed sets.[3][4]
A map may be open, closed, both, or neither;[5] in particular, an open map need not be closed and vice versa.[6]
Open[7] and closed[8] maps are not necessarily continuous.[4] Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;[3] this fact remains true even if one restricts oneself to metric spaces.[9]
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 preimage of every open set of
is open in
[2] (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.[10]