Editing Open and closed maps (section)

{{Short description|A function that sends open (resp. closed) subsets to open (resp. closed) subsets}}
In [[mathematics]], more specifically in [[topology]], an '''open map''' is a [[function (mathematics)|function]] between two [[topological space]]s that maps [[open set]]s to open sets.<ref>{{cite book|last=Munkres|first=James R.|author-link=James Munkres|title=Topology|edition=2nd|publisher=[[Prentice Hall]]|year=2000|isbn=0-13-181629-2}}</ref><ref name=mendelson>{{cite book|last=Mendelson|first=Bert|date=1990|orig-year=1975|title=Introduction to Topology|edition=Third|publisher=Dover|isbn=0-486-66352-3|page=89|quote=It is important to remember that Theorem 5.3 says that a function <math>f</math> is continuous if and only if the {{em|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 {{em|open mappings}}).}}</ref><ref name=lee550>{{cite book|last=Lee|first=John M.|date=2003|title=Introduction to Smooth Manifolds|series=Graduate Texts in Mathematics|volume=218|publisher=Springer Science & Business Media|isbn=9780387954486|page=550|quote=A map <math>F : X \to Y</math> (continuous or not) is said to be an {{em|open map}} if for every closed subset <math>U \subseteq X,</math> <math>F(U)</math> is open in <math>Y,</math> and a {{em|closed map}} if for every closed subset <math>K \subseteq U,</math> <math>F(K)</math> is closed in <math>Y.</math> Continuous maps may be open, closed, both, or neither, as can be seen by examining simple examples involving subsets of the plane.}}</ref> 
That is, a function <math>f : X \to Y</math> is open if for any open set <math>U</math> in <math>X,</math> the [[Image (mathematics)|image]] <math>f(U)</math> is open in <math>Y.</math> 
Likewise, a '''closed map''' is a function that maps [[closed set]]s to closed sets.<ref name=lee550/><ref name=ludu15>{{cite book|last=Ludu|first=Andrei|title=Nonlinear Waves and Solitons on Contours and Closed Surfaces|series=Springer Series in Synergetics|date=15 January 2012|isbn=9783642228940|page=15|quote=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.}}</ref> 
A map may be open, closed, both, or neither;<ref>{{cite book|last=Sohrab|first=Houshang H.|date=2003|title=Basic Real Analysis|publisher=Springer Science & Business Media|isbn=9780817642112|url=https://books.google.com/books?id=QnpqBQAAQBAJ&pg=PA203|page=203|quote=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.)</ref> in particular, an open map need not be closed and vice versa.<ref>{{cite book|last=Naber|first=Gregory L.|date=2012|title=Topological Methods in Euclidean Spaces|edition=reprint|series=Dover Books on Mathematics|publisher=Courier Corporation|isbn=9780486153445|page=18|quote=''Exercise 1-19.'' Show that the projection map <math>\pi_i : X_i \times \cdots \times X_k \to X_i</math>π<sub>1</sub>:''X''<sub>1</sub> × ··· × ''X''<sub>''k''</sub> → ''X''<sub>i</sub> is an open map, but need not be a closed map. Hint: The projection of '''R'''<sup>2</sup> onto <math>\R</math> 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.}}</ref>

Open<ref name=mendelson2>{{cite book|last=Mendelson|first=Bert|date=1990|orig-year=1975|title=Introduction to Topology|edition=Third|publisher=Dover|isbn=0-486-66352-3|page=89|quote=There are many situations in which a function <math>f : \left( X, \tau\right) \to \left( Y, \tau' \right)</math> has the property that for each open subset <math>A</math> of <math>X,</math> the set <math>f(A)</math> is an open subset of <math>Y,</math> and yet <math>f</math> is {{em|not}} continuous.}}</ref> and closed<ref>{{cite book|last=Boos|first=Johann|date=2000|title=Classical and Modern Methods in Summability|series=Oxford University Press|isbn=0-19-850165-X|url=https://books.google.com/books?id=kZ9cy6XyidEC&pg=PA332|page=332|quote=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.}}</ref> maps are not necessarily [[Continuous function (topology)|continuous]].<ref name=ludu15/> Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;<ref name=lee550/> this fact remains true even if one restricts oneself to metric spaces.<ref>{{cite book|last=Kubrusly|first=Carlos S.|date=2011|title=The Elements of Operator Theory|url=https://archive.org/details/elementsoperator00kubr|url-access=limited|publisher=Springer Science & Business Media|isbn=9780817649982|page=[https://archive.org/details/elementsoperator00kubr/page/n131 115]|quote=In general, a map <math>F : X \to Y</math> of a metric space <math>X</math> into a metric space <math>Y</math> may possess any combination of the attributes 'continuous', 'open', and 'closed' (that is, these are independent concepts).}}</ref> 
Although their definitions seem more natural, open and closed maps are much less important than continuous maps. 
Recall that, by definition, a function <math>f : X \to Y</math> is continuous if the [[preimage]] of every open set of <math>Y</math> is open in <math>X.</math><ref name=mendelson/> (Equivalently, if the preimage of every closed set of <math>Y</math> is closed in <math>X</math>).

Early study of open maps was pioneered by [[Simion Stoilow]] and [[Gordon Thomas Whyburn]].<ref>{{cite book|editor1-last=Hart|editor1-first=K. P.|editor2-last=Nagata|editor2-first=J.|editor3-last=Vaughan|editor3-first=J. E.|date=2004|title=Encyclopedia of General Topology|url=https://archive.org/details/encyclopediagene00hart_882|url-access=limited|publisher=Elsevier|isbn=0-444-50355-2|page=[https://archive.org/details/encyclopediagene00hart_882/page/n96 86]|quote=It seems that the study of open (interior) maps began with papers [13,14] by [[Simion Stoilow|S. Stoïlow]]. Clearly, openness of maps was first studied extensively by [[Gordon Thomas Whyburn|G.T. Whyburn]] [19,20].}}</ref>