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

Definitions and characterizations

If is a subset of a topological space then let and denote the closure of in that space.
Let be a function between topological spaces. If is any set then is called the image of under

Competing definitions

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 image 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.
A surjective map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.
More generally, a map is relatively open if and only if the surjection is a strongly open map.
Because is always an open subset of the image of a strongly open map must be an open subset of its codomain In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain.
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".

Open maps

A map is called an ' or a ' if it satisfies any of the following equivalent conditions:

  1. Definition: maps open subsets of its domain to open subsets of its codomain; that is, for any open subset of, is an open subset of
  2. is a relatively open map and its image is an open subset of its codomain
  3. For every and every neighborhood of , is a neighborhood of. We can replace the first or both instances of the word "neighborhood" with "open neighborhood" in this condition and the result will still be an equivalent condition:
    • For every and every open neighborhood of, is a neighborhood of.
    • For every and every open neighborhood of, is an open neighborhood of.
  4. for all subsets of where denotes the topological interior of the set.
  5. Whenever is a closed subset of then the set is a closed subset of
    • This is a consequence of the identity which holds for all subsets
If is a basis for then the following can be appended to this list:
  2. maps basic open sets to open sets in its codomain.

    3. Closed maps

A map is called a ' if whenever is a closed subset of the domain then is a closed subset of 's image where as usual, this set is endowed with the subspace topology induced on it by 's codomain
A map is called a ' or a if it satisfies any of the following equivalent conditions:

  1. Definition: maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset of is a closed subset of
  2. is a relatively closed map and its image is a closed subset of its codomain
  3. for every subset
  4. for every closed subset
  5. Whenever is an open subset of then the set is an open subset of
  6. If is a net in and is a point such that in then converges in to the set
    • The convergence means that every open subset of that contains will contain for all sufficiently large indices
A surjective map is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent.
By definition, the map is a relatively closed map if and only if the surjection is a strongly closed map.
If in the open set definition of "continuous map", both instances of the word "open" are replaced with "closed" then the statement of results is to continuity.
This does not happen with the definition of "open map" since the statement that results is the definition of "closed map", which is in general equivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any set only is guaranteed in general, whereas for preimages, equality always holds.

Examples

The function defined by is continuous, closed, and relatively open, but not open. This is because if is any open interval in 's domain that does contain then where this open interval is an open subset of both and However, if is any open interval in that contains then which is not an open subset of 's codomain but an open subset of Because the set of all open intervals in is a basis for the Euclidean topology on this shows that is relatively open but not open.
If has the discrete topology then every function is both open and closed.
For example, the floor function from Real number| to Integer| 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 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.

Sufficient conditions

Every [homeomorphism is open, closed, and continuous. In fact, a continuous bijection is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.
The composition of two open maps is an open map and the composition of two closed maps is a closed map. However, the composition of two relatively open maps need not be relatively open and the composition of two relatively closed maps need not be relatively closed.
If is strongly open and is relatively open, then is relatively open.
Let be a map. Given any subset, if is relatively open, then the same is true of its restriction
to the -saturated subset.
The categorical sum of two open maps is open, and of two closed maps is closed. The categorical product of two open maps is also 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 continuous bijection is an open and closed bijection.
An open surjection is not necessarily closed, and a closed surjection is not necessarily open. All local homeomorphisms, including all coordinate charts on manifolds and all covering maps, are open maps.
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 continuous linear surjection between Banach spaces is an open map. This theorem has been generalized to topological vector spaces beyond just Banach spaces.
A surjective map is called an ' if, for every, there exists some such that is a ' for which by definition means that, for every open neighborhood of, is a neighborhood of in .
Every open surjection is an almost open map, but the converse is false.
If a surjection is an almost open map, then it is an open map if it satisfies the following condition : whenever and belong to the same fiber of , then, for every neighborhood of, there exists some neighborhood of such that.
If the map is continuous, then the above condition is also necessary for the map to be open. That is, if is a continuous surjection, then it is open if and only if it is almost open and it satisfies the above condition.