[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.
is continuous if the preimage of every open set of
[2] (Equivalently, if the preimage of every closed set of
Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.
There are two different competing, but closely related, definitions of "open map" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets."
A surjective map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.
In summary, By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition.
The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".
is called a relatively closed map if whenever
where as usual, this set is endowed with the subspace topology induced on it by
So for this important special case, the two definitions are equivalent.
If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") is equivalent to continuity.
This does not happen with the definition of "open map" (which is: "every image of an open set is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in general not equivalent to openness.
has the discrete topology (that is, all subsets are open and closed) then every function
Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps.
To every point on the unit circle we can associate the angle of the positive
-axis with the ray connecting the point with the origin.
This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous.
Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed.
In fact, a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.
is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, surjective) map then the same is true of its restriction
[15] The categorical product of two open maps is open, however, the categorical product of two closed maps need not be closed.
All local homeomorphisms, including all coordinate charts on manifolds and all covering maps, are open maps.
Closed map lemma — Every continuous function
A variant of the closed map lemma states that if a continuous function between locally compact Hausdorff spaces is proper then it is also closed.
In complex analysis, the identically named open mapping theorem states that every non-constant holomorphic function defined on a connected open subset of the complex plane is an open map.
The invariance of domain theorem states that a continuous and locally injective function between two
In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map.
is a continuous map that is also open or closed then: In the first two cases, being open or closed is merely a sufficient condition for the conclusion that follows.