Open and closed maps


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 image 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 preimage of every open set of is open in X..
Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.

Definition and characterizations

Let be a function between topological spaces.

Open maps

We say that is an open map if it satisfies any of the following equivalent conditions:

  1. maps open sets to open sets ;
  2. for every and every neighborhood of , there exists a neighborhood of such that ;
  3. for all subsets of, where denotes the topological interior of the set;
  4. whenever is a closed subset of then the set is closed in.
and if is a basis for then we may add to this list:

  1. maps basic open sets to open sets ;
We say that is a relatively open map if is an open map, where is the range or image of.

Closed maps

We say that is a closed map if it satisfies any of the following equivalent conditions:

  1. maps closed sets to closed sets ;
  2. for all subsets of.
We say that is a relatively closed map if is a closed map.

Sufficient conditions

The composition of two open maps is again open; the composition of two closed maps is again closed.
The categorical sum of two open maps is open, or of two closed maps is closed.
The categorical product of two open maps is open, however, the categorical product of two closed maps need not be closed.
A bijective map is open if and only if it is closed.
The inverse of a bijective continuous map is a bijective open/closed map.
A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map.
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 -dimensional topological manifolds must be open.
In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map.
This theorem has been generalized to topological vector spaces beyond just Banach spaces.

Examples

Every homeomorphism is open, closed, and continuous. 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.
If Y has the discrete topology then every function is both open and closed.
For example, the floor function from R to Z is open and closed, but not continuous.
This example shows that the image of a connected space under an open or closed map need not be connected.
Whenever we have a product of topological spaces, the natural projections are open.
Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps.
Projections need not be closed however. Consider for instance the projection on the first component; then the set is closed in, but is not closed in.
However, for a compact space Y, the projection is closed. This is essentially the tube lemma.
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.
It shows that the image of a compact space under an open or closed map need not be compact.
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. Specifying the codomain is essential.
The function f : R → R with f = x2 is continuous and closed, but not open.

Properties

Let be a continuous map that is either open or closed.
Then
In the first two cases, being open or closed is merely a sufficient condition for the result to follow.
In the third case, it is necessary as well.