Open and closed maps

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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]

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  

Completing definitionsEdit

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  [11]

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".

Open mapsEdit

A map   is called an open map or a strongly open map 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   (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.
  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  [12]

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

  1.   maps basic open sets to open sets in its codomain (that is, for any basic open set     is an open subset of  ).

Closed mapsEdit

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:

  1. 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  
  2.   is a relatively closed map and its image   is a closed subset of its codomain  
  3.   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.

ExamplesEdit

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[13][14] (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.

Sufficient conditionsEdit

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 open maps (resp. closed maps)   and   is again an open map (resp. a closed map)  [15][16] If however,   is not an open (resp. closed) subset of   then this is no longer guaranteed.

The categorical sum of two open maps is open, or of two closed maps is closed.[16] The categorical product of two open maps is open, however, the categorical product of two closed maps need not be closed.[15][16]

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.

Closed map lemma — Every continuous function   from a compact space   to a Hausdorff space   is closed and proper (i.e. 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 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.

PropertiesEdit

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.

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 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
     
    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   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  

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.

See alsoEdit

  • 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

NotesEdit

  1. ^ a b 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  
  2. ^ 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  

CitationsEdit

  1. ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
  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).
  3. ^ a b c 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.
  4. ^ a b 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.
  5. ^ 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.)
  6. ^ 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 × ··· × XkXi 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ 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).
  10. ^ 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].
  11. ^ Narici & Beckenstein 2011, pp. 225-273.
  12. ^ Stack exchange post
  13. ^ Willard, Stephen (1970). General Topology. Addison-Wesley. ISBN 0486131785.
  14. ^ 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.
  15. ^ a b 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,...
  16. ^ 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.

ReferencesEdit