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In mathematics, more specifically in topology, an **open map** is a function between two topological spaces that maps open sets to open sets.^{[1]}^{[2]}^{[3]}
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.^{[3]}^{[4]}
A map may be open, closed, both, or neither;^{[5]} in particular, an open map need not be closed and vice versa.^{[6]}

Open^{[7]} and closed^{[8]} maps are not necessarily continuous.^{[4]} Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;^{[3]} this fact remains true even if one restricts oneself to metric spaces.^{[9]}
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 ^{[2]} (Equivalently, if the preimage of every closed set of is closed in ).

Early study of open maps was pioneered by Simion Stoilow and Gordon Thomas Whyburn.^{[10]}

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

- "
" if whenever is an open subset of the domain then is an open subset of 's codomain*Strongly 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*Relatively open map*^{[11]}

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

**Warning**: Many authors define "open map" to mean "*relatively*open map" (for example, 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.

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,

- 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".

A map is called an ** open map** or a

- 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), 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 .

- 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
- 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:

- 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

- 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
- is a relatively closed map and its image is a closed subset of its codomain
- for every subset
- for every closed subset
- for every closed subset
- Whenever is an open subset of then the set is an open subset of
- 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" (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 of results ("every preimage of a closed set is closed") is *equivalent* 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 *not* 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.

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 (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 ** ** 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^{[12]}^{[13]} (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.

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 composition of two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map.^{[14]}^{[15]} 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) map then the same is true of its restriction to the -saturated subset

The categorical sum of two open maps is open, or of two closed maps is closed.^{[15]}
The categorical product of two open maps is open, however, the categorical product of two closed maps need not be closed.^{[14]}^{[15]}

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.

**Closed map lemma** — Every continuous function from a compact space to a Hausdorff space is closed and proper (meaning that preimages of compact sets are compact).

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.

**Invariance of domain** — If is an open subset of and is an injective continuous map, then is open in and is a homeomorphism between and

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

- whenever belong to the same 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.

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.

- A surjective map is called
- if is an 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 as well.

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
where this set is also necessarily a regular closed set (in ).
^{[note 1]}In particular, if is a regular closed set then so is And if is a regular open set then so is - 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 - If a net converges in to a point and if the continuous open map is surjective, then for any there exists a net in (indexed by some directed set ) such that in and is a subnet of Moreover, the indexing set may be taken to be with the product order where is any neighbourhood basis of directed by
^{[note 2]}

- Almost open map – Map that satisfies a condition similar to that of being an open map.
- Closed graph – Graph of a map closed in the product space
- Closed linear operator
- Local homeomorphism – Mathematical function revertible near each point
- Quasi-open map – Function that maps non-empty open sets to sets that have non-empty interior in its codomain
- Quotient map (topology) – Topological space construction
- Perfect map – Continuous closed surjective map, each of whose fibers are also compact sets
- Proper map – Map between topological spaces with the property that the preimage of every compact is compact
- Sequence covering map

- ^
^{a}^{b}A subset is called aif or equivalently, if where (resp. ) denotes the topological boundary (resp. interior, closure) of in The set is called a*regular closed 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*regular open set* **^**Explicitly, for any pick any such that and then let be arbitrary. The assignment defines an order morphism such that is a cofinal subset of thus is a Willard-subnet of

**^**Munkres, James R. (2000).*Topology*(2nd ed.). Prentice Hall. ISBN 0-13-181629-2.- ^
^{a}^{b}Mendelson, Bert (1990) [1975].*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*). - ^
^{a}^{b}^{c}Lee, John M. (2003).*Introduction to Smooth Manifolds*. Graduate Texts in Mathematics. Vol. 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. - ^
^{a}^{b}Ludu, Andrei (15 January 2012).*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 π_{1}:*X*_{1}× ··· ×*X*_{k}→*X*_{i}is an open map, but need not be a closed map. Hint: The projection of**R**^{2}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) [1975].*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' (that is, 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.**^**Willard, Stephen (1970).*General Topology*. Addison-Wesley. ISBN 0486131785.**^**Lee, John M. (2012).*Introduction to Smooth Manifolds*. Graduate Texts in Mathematics. Vol. 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.- ^
^{a}^{b}Baues, Hans-Joachim; Quintero, Antonio (2001).*Infinite Homotopy Theory*.*K*-Monographs in Mathematics. Vol. 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,...

- ^
^{a}^{b}^{c}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.

- Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.