In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. the image 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. is continuous if and only if the preimage).

Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.

Definitions and characterizations

) denote the closure (resp. interior be a function between topological spaces

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.

is called a

• " is an open subset's codomain

• "'s image where as usual, this set is endowed with the subspace topology

Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.

1. Warning: Many authors define "open map" to mean " open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean " open map". In general, these definitions are equivalent so it is thus advisable to always check what definition of "open map" an author is using.

A surjective is relatively open if and only if the surjection is a strongly open map.

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,

1. A map is strongly open if and only if it is relatively open and its image is an open subset of its codomain.

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

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

and every neighborhood. 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:

• .

• .

denotes the topological interior of the set. is a closed subset

• This is a consequence of the identity

is a basis then the following can be appended to this list:

1. ).

Closed maps

is called a is a closed subset's image where as usual, this set is endowed with the subspace topology's codomain

is called a or a if it satisfies any of the following equivalent conditions:

is a net

A surjective 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" (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 that results ("every preimage of a closed set is closed") is 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 always holds.

Examples

that does but is a basis for the Euclidean topology is relatively open but not (strongly) open.

has the discrete topology is both open and closed (but not necessarily continuous). 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 are open (as well as continuous). Since the projections of fiber bundles and covering maps is closed. This is essentially the tube lemma.

To every point on the unit circle we can associate the angle-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval 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 (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map. is relatively open (respectively, relatively closed).

is relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, surjective to the f-saturated.

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 (and vice versa). 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-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.

is called an is a is a neighborhood is not required to be an is an almost open map, then it is an open map if it satisfies the following condition (a condition that does belong to the same fiber is a continuous surjection, then it is open if and only if it is almost open and it satisfies the above condition.

Properties

Open or closed maps that are continuous

is a continuous map that is also open closed, then:

is a surjection, then it is a quotient map and even a hereditarily quotient map,

• is called is a quotient map.

• is an injection, then it is a topological embedding.

• is a bijection, then it is a homeomorphism.

In the first two cases, being open or closed is merely a sufficient condition for the conclusion that follows. In the third case, it is necessary as well.

Open continuous maps

then:

denotes the boundary of a set. denote the closure of a set. denotes the interior is also necessarily a regular closed set). is a regular open set is a regular open (resp. a regular closed). If a net converges (indexed by some directed set is a subnet with the product order is any neighbourhood basis

See also

Notes

Citations

References