Definitions and characterizations
If is a subset of a topological space then let and (resp. ) denote the Closure (topology), closure (resp. Interior (topology), interior) of in that space. Let be a function between topological spaces. If is any set then is called the image of underCompeting 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 set, 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 (mathematics), image where as usual, this set is endowed with the subspace topology induced on it by 's codomain A Surjective function, surjective map is relatively open if and only if it strongly open; so for this important special case the definitions are equivalent. More generally, the map is a relatively open map if and only if the Surjective function, surjection is a strongly open map. :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. Every strongly open map is a relatively open map. And because is always an open subset of the image of a strongly open map must be an open subset of However, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain In summary, :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
A map is called an or a if it satisfies any of the following equivalent conditions:Closed maps
A map is called a if whenever is a Closed set, closed subset of the domain then is a closed subset of 's Image (mathematics), 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:Examples
The function defined by is continuous, closed, and relatively open, but not (strongly) 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 (topology), basis for the Euclidean topology on this shows that is relatively open but not (strongly) open. If has the discrete topology (that is, all subsets are open and closed) then every function is both open and closed (but not necessarily continuous). 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 topology, product of topological spaces the natural projections are open (as well as continuous). 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 (mathematics), intervalSufficient conditions
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. The Function composition, composition of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map. However, the composition of two relatively open maps need not be relatively open and similarly, the composition of two relatively closed maps need not be relatively closed. If is strongly open (respectively, strongly closed) and is relatively open (respectively, relatively closed) then is relatively open (respectively, relatively closed). Let be a map. Given any subset if is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous, Surjective function, surjective) map then the same is true of its restriction to the Saturated set, -saturated subset The categorical sum of two open maps is open, or of two closed maps is closed. The categorical Product (topology), 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 (and vice versa). A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. 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 space, locally compact Hausdorff spaces is proper then it is also closed. In complex analysis, the identically named Open mapping theorem (complex analysis), open mapping theorem states that every non-constant holomorphic function defined on a Connected space, 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 Manifold, topological manifolds must be open. In functional analysis, the Open mapping theorem (functional analysis), 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. 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 (topology), neighborhood of in (note that the neighborhood is not required to be an neighborhood). Every surjective open map is an almost open map but in general, the converse is not necessarily true. If a surjection is an almost open map then it will be an open map if it satisfies the following condition (a condition that does depend in any way on 's topology ): :whenever belong to the same Fiber (mathematics), fiber of (that is, ) 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 an open map if and only if it is almost open and it satisfies the above condition.Properties
Open or closed maps that are continuous
If is a continuous map that is also open closed then: * if is a surjection then it is a quotient map and even a hereditarily quotient map, ** A surjective map is called if for every subset the restriction is a quotient map. * if is an Injective function, injection then it is a topological embedding. * if 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 condition, necessary as well.Open continuous maps
If is a continuous (strongly) open map, and then:See also
* * * * * * * * *Notes
Citations
References
* * * {{DEFAULTSORT:Open And Closed Maps General topology Continuous mappings Lemmas