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  (Equivalently, if the preimage of every closed set of is closed in ).
Definitions and characterizationsEdit
If is a subset of a topological space then let and (resp. ) denote the closure (resp. interior) of in that space. Let be a function between topological spaces. If is any set then is called the image of under
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." The following terminology is sometimes used to distinguish between the two definitions.
A map is called a
- "Strongly open map" if whenever is an open subset of the domain then is an open subset of 's codomain
- "Relatively open map" 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 
A surjective map is relatively open if and only if it a 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 surjection is a strongly open map.
- Warning: Many authors define "open map" to mean "relatively open map" (e.g. The Encyclopedia of Mathematics) while others define "open map" to mean "strongly open map". In general, these definitions are not 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. This subsection's discussion will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".
A map is called an open map or a strongly open map if it satisfies any of the following equivalent conditions:
- 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
- is a relatively open map and its image is an open subset of its codomain .
- For every and every neighborhood of (however small), there exists a neighborhood of such that .
- Either instance of the word "neighborhood" in this statement can be replaced with "open neighborhood" and the resulting statement would still characterize strongly open maps.
- for all subsets of where denotes the topological interior of the set.
- Whenever is a closed subset of then the set is a closed subset of 
and if is a basis for then the following can be appended to this list:
- maps basic open sets to open sets in its codomain (that is, for any basic open set is an open subset of ).
A map is called a relatively closed map 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 closed map or a strongly closed map if it satisfies any of the following equivalent conditions:
- Definition: maps closed subsets of its domain to closed subsets of its codomain; that is, for any closed subset of is an closed subset of
- is a relatively closed map and its image is a closed subset of its codomain
- for every subset
A surjective map is strongly closed if and only if it a 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.
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 not 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 is 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 (strongly) open.
If has the discrete topology (i.e. all subsets are open and closed) then every function is both open and closed (but not necessarily continuous). For example, the floor function from to 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 (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 [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 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 (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.
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 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.
A surjective map is called an almost open map if for every there exists some such that is a point of openness for which by definition means that for every open neighborhood of is a neighborhood of in (note that the neighborhood is not required to be an open 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 not depend in any way on 's topology ):
- whenever belong to the same fiber of (i.e. ) 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.
Let be a map. Given any subset if is a relatively open (resp. relatively closed, strongly open, strongly closed, continuous, surjective) map then the same is true of its restriction
to the -saturated subset
Open or closed maps that are continuousEdit
If is a continuous map that is also open or closed then:
- if is a surjection then it is a quotient map and even a hereditarily quotient map,
- A surjective map is called hereditarily quotient if for every subset the restriction is a quotient map.
- if is an injection then it is a topological embedding.
- if is a bijection then it is a homeomorphism.
Open continuous mapsEdit
If is a continuous (strongly) open map, and then:
- where denotes the boundary of a set.
- where denote the closure of a set.
- If where denotes the interior of a set, then
- If the continuous open map is also surjective then and moreover, is a regular open (resp. a regular closed)[note 1] subset of if and only if is a regular open (resp. a regular closed) subset of
Suppose is a function and is a surjective map. There might not exist any map such that on This motivates defining the set which will be the largest subset of on which such a function may be defined unambiguously (without resorting to arbitrarily picking a value). Explicitly, let be the set of all such that the restriction of to the fiber is a constant map (or equivalently, such that is a singleton set). For any such let denote the constant value that takes on the fiber This induces a map which is the unique map satisfying for every
The importance of this map is that holds on where by its very definition, the set is the (unique) largest subset of on which such a map may be defined. If is a continuous open surjection from a first-countable space onto a Hausdorff space and if is a continuous map valued in a Hausdorff space then is a closed subset of [note 2] the surjection is continuous and open, and (as a consequence of holding on ) the map is continuous. The set being closed in obviates any possible need to continuously extend to the closure (in ) of its domain.
- Almost open map – A map that satisfies a condition similar to that of being an open map.
- Closed graph – a graph of a function that is also a closed subset of the product space
- Closed linear operator
- Local homeomorphism – A continuous open map that, around every point in its domain, has a neighborhood on which it restricts to a homomorphism
- Quasi-open map – A function that maps non-empty open sets to sets that have non-empty interior in its codomain.
- Quotient map
- Perfect map – A continuous closed surjective map, each of whose fibers are also compact sets.
- Proper map – A map between topological spaces with the property that the preimage of every compact is compact
- Sequence covering map
- A subset is called a regular closed set if or equivalently, if where (resp. ) denotes the topological boundary (resp. interior, closure) of in The set is called a regular open set if or equivalently, if The interior (taken in ) of a closed subset of is always a regular open subset of The closure (taken in ) of an open subset of is always a regular closed subset of
- The less trivial conclusion that is always a closed subset of was reached despite the fact that the definition of is purely set-theoretic and not in any way dependent on any topology (although the requirement that be continuous limits which functions of the form are considered, it does not influence the definition of ). Moreover, this result shows that for every Hausdorff space and every continuous map (where this space and map are chosen without regard to and ) the set is nevertheless necessarily closed in
- Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
- Mendelson, Bert (1990) . Introduction to Topology (Third ed.). Dover. p. 89. ISBN 0-486-66352-3.
It is important to remember that Theorem 5.3 says that a function is continuous if and only if the inverse image of each open set is open. This characterization of continuity should not be confused with another property that a function may or may not possess, the property that the image of each open set is an open set (such functions are called open mappings).
- Lee, John M. (2003). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218. Springer Science & Business Media. p. 550. ISBN 9780387954486.
A map (continuous or not) is said to be an open map if for every closed subset is open in and a closed map if for every closed subset is closed in Continuous maps may be open, closed, both, or neither, as can be seen by examining simple examples involving subsets of the plane.
- Ludu, Andrei. Nonlinear Waves and Solitons on Contours and Closed Surfaces. Springer Series in Synergetics. p. 15. ISBN 9783642228940.
An open map is a function between two topological spaces which maps open sets to open sets. Likewise, a closed map is a function which maps closed sets to closed sets. The open or closed maps are not necessarily continuous.
- Sohrab, Houshang H. (2003). Basic Real Analysis. Springer Science & Business Media. p. 203. ISBN 9780817642112.
Now we are ready for our examples which show that a function may be open without being closed or closed without being open. Also, a function may be simultaneously open and closed or neither open nor closed.(The quoted statement in given in the context of metric spaces but as topological spaces arise as generalizations of metric spaces, the statement holds there as well.)
- Naber, Gregory L. (2012). Topological Methods in Euclidean Spaces. Dover Books on Mathematics (reprint ed.). Courier Corporation. p. 18. ISBN 9780486153445.
Exercise 1-19. Show that the projection map π1:X1 × ··· × Xk → Xi is an open map, but need not be a closed map. Hint: The projection of R2 onto is not closed. Similarly, a closed map need not be open since any constant map is closed. For maps that are one-to-one and onto, however, the concepts of 'open' and 'closed' are equivalent.
- Mendelson, Bert (1990) . Introduction to Topology (Third ed.). Dover. p. 89. ISBN 0-486-66352-3.
There are many situations in which a function has the property that for each open subset of the set is an open subset of and yet is not continuous.
- Boos, Johann (2000). Classical and Modern Methods in Summability. Oxford University Press. p. 332. ISBN 0-19-850165-X.
Now, the question arises whether the last statement is true in general, that is whether closed maps are continuous. That fails in general as the following example proves.
- Kubrusly, Carlos S. (2011). The Elements of Operator Theory. Springer Science & Business Media. p. 115. ISBN 9780817649982.
In general, a map of a metric space into a metric space may possess any combination of the attributes 'continuous', 'open', and 'closed' (i.e., these are independent concepts).
- Hart, K. P.; Nagata, J.; Vaughan, J. E., eds. (2004). Encyclopedia of General Topology. Elsevier. p. 86. ISBN 0-444-50355-2.
It seems that the study of open (interior) maps began with papers [13,14] by S. Stoïlow. Clearly, openness of maps was first studied extensively by G.T. Whyburn [19,20].
- Narici & Beckenstein 2011, pp. 225-273.
- Stack exchange post
- Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 0486131785.
- Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. 218 (Second ed.). p. 606. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9982-5.
Exercise A.32. Suppose are topological spaces. Show that each projection is an open map.
- Baues, Hans-Joachim; Quintero, Antonio (2001). Infinite Homotopy Theory. K-Monographs in Mathematics. 6. p. 53. ISBN 9780792369820.
A composite of open maps is open and a composite of closed maps is closed. Also, a product of open maps is open. In contrast, a product of closed maps is not necessarily closed,...
- James, I. M. (1984). General Topology and Homotopy Theory. Springer-Verlag. p. 49. ISBN 9781461382836.
...let us recall that the composition of open maps is open and the composition of closed maps is closed. Also that the sum of open maps is open and the sum of closed maps is closed. However, the product of closed maps is not necessarily closed, although the product of open maps is open.
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