closed neighborhoods

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



f
:
X
→
Y


{\displaystyle f:X\to Y}
is open if for any open set



U


{\displaystyle U}
in



X
,


{\displaystyle X,}
the image



f
(
U
)


{\displaystyle f(U)}
is open in



Y
.


{\displaystyle Y.}

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



f
:
X
→
Y


{\displaystyle f:X\to Y}
is continuous if the preimage of every open set of



Y


{\displaystyle Y}
is open in



X
.


{\displaystyle X.}
(Equivalently, if the preimage of every closed set of



Y


{\displaystyle Y}
is closed in



X


{\displaystyle X}
).
Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.

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