Editing Pointless topology

{{Short description|Mathematical approach}}
In [[mathematics]], '''pointless topology''', also called '''point-free topology''' (or '''pointfree topology''') or '''topology without points''' and '''locale theory''', is an approach to [[topology]] that avoids mentioning [[point (mathematics)|points]], and in which the [[Lattice (order)|lattices]] of [[Open set|open sets]] are the primitive notions.{{sfn|Johnstone|1983|p=41}} In this approach it becomes possible to construct ''topologically interesting'' spaces from purely algebraic data.{{sfn|Johnstone|1983|p=42}} 

== History ==
The first approaches to topology were geometrical, where one started from [[Euclidean space]] and patched things together. But [[Marshall Stone]]'s work on [[Stone duality]] in the 1930s showed that topology can be viewed from an algebraic point of view (lattice-theoretic). Apart from Stone, [[Henry Wallman]] was the first person to exploit this idea. Others continued this path till [[Charles Ehresmann]] and his student [[Jean Bénabou]] (and simultaneously others), made the next fundamental step in the late fifties. Their insights arose from the study of "topological" and "differentiable" [[Category (mathematics)|categories]].{{sfn|Johnstone|1983|p=42}}

Ehresmann's approach involved using a category whose objects were [[Complete lattice|complete lattices]] which satisfied a [[Distributive property|distributive]] law and whose [[morphism]]s were maps which preserved finite [[Join and meet|meets]] and arbitrary [[Join and meet|joins]]. He called such lattices "local lattices"; today they are called "frames" to avoid ambiguity with other notions in [[lattice theory]].{{sfn|Johnstone|1983|p=43}}

The theory of [[frames and locales]] in the contemporary sense was developed through the following decades ([[John R. Isbell|John Isbell]], [[Peter Johnstone (mathematician)|Peter Johnstone]], Harold Simmons, [[:de:Bernhard Banaschewski|Bernhard Banaschewski]], [[:cs:Aleš Pultr|Aleš Pultr]], Till Plewe, Japie Vermeulen, [[Steve Vickers (computer scientist)|Steve Vickers]]) into a lively branch of topology, with application in various fields, in particular also in theoretical computer science. For more on the history of locale theory see Johnstone's overview.<ref>Peter T. Johnstone, Elements of the history of locale theory, in: Handbook of the History of General Topology, vol. 3, pp. 835-851, Springer, {{ISBN|978-0-7923-6970-7}}, 2001.</ref>

==Intuition==
Traditionally, a [[topological space]] consists of a [[Set (mathematics)|set]] of [[point (topology)|points]] together with a ''topology'', a system of subsets called [[open set]]s that with the operations of  [[union (set theory)|union]] (as [[Join (mathematics)|join]]) and [[intersection (set theory)|intersection]] (as  [[Meet (Mathematics)|meet]]) forms a [[lattice (order)|lattice]] with certain properties. Specifically, the union of any family of open sets is again an open set, and the intersection of finitely many open sets is again open. In pointless topology we take these properties of the lattice as fundamental, without requiring that the lattice elements be sets of points of some underlying space and that the lattice operation be intersection and union. Rather, point-free topology is based on the concept of a "realistic spot" instead of a point without extent. These "spots" can be [[Join (mathematics)|joined]] (symbol <math>\vee </math>), akin to a union, and we also have a [[Meet (Mathematics)|meet]] operation for spots (symbol <math>\and </math>), akin to an intersection. Using these two operations, the spots form a [[complete lattice]]. If a spot meets a join of others it has to meet some of the constituents, which, roughly speaking, leads to the distributive law

:<math>b \wedge \left( \bigvee_{i\in I} a_i\right) = \bigvee_{i\in I} \left(b\wedge a_i\right)</math>
where the <math>a_i</math> and <math>b</math> are spots and the index family <math>I</math> can be arbitrarily large. This distributive law is also satisfied by the lattice of open sets of a topological space.

If <math>X</math> and <math>Y</math> are topological spaces with lattices of open sets denoted by <math>\Omega(X)</math> and <math>\Omega(Y)</math>, respectively, and <math>f\colon X\to Y</math> is a [[Continuous function|continuous map]], then, since the [[pre-image]] of an open set under a continuous map is open, we obtain a map of lattices in the opposite direction: <math>f^*\colon \Omega(Y)\to \Omega(X)</math>. Such "opposite-direction" lattice maps thus serve as the proper generalization of continuous maps in the point-free setting.

==Formal definitions==

The basic concept is that of a '''frame''', a [[complete lattice]] satisfying the general distributive law above. '''Frame homomorphisms''' are maps between frames that respect all [[Join (mathematics)|joins]] (in particular, the [[least element]] of the lattice) and finite [[meet (mathematics)|meet]]s (in particular, the [[greatest element]] of the lattice). Frames, together with frame homomorphisms, form a [[category (mathematics)|category]].

The [[opposite category]] of the category of frames is known as the '''category of locales'''. A locale <math>X</math> is thus nothing but a frame; if we consider it as a frame, we will write it as <math>O(X)</math>. A '''locale morphism''' <math>X\to Y</math> from the locale <math>X</math> to the locale <math>Y</math> is given by a frame homomorphism <math>O(Y)\to O(X)</math>.

Every topological space <math>T</math> gives rise to a frame <math>\Omega(T)</math> of open sets and thus to a locale. A locale is called '''spatial''' if it isomorphic (in the category of locales) to a locale arising from a topological space in this manner.

== Examples of locales ==

* As mentioned above, every topological space <math>T</math> gives rise to a frame <math>\Omega(T)</math> of open sets and thus to a locale, by definition a spatial one.
* Given a topological space <math>T</math>, we can also consider the collection of its [[Regular open set|regular open sets]]. This is a frame using as join the interior of the closure of the union, and as meet the intersection. We thus obtain another locale associated to <math>T</math>. This locale will usually not be spatial.
* For each <math>n\in\N</math> and each <math>a\in\R</math>, use a symbol <math>U_{n,a}</math> and construct the free frame on these symbols, modulo the relations

::<math>\bigvee_{a\in\R} U_{n,a}=\top   \ \text{  for every }n\in\N</math>

::<math>U_{n,a}\and U_{n,b}=\bot   \ \text{  for every }n\in\N\text{ and all }a,b\in\R\text{ with } a\ne b</math>

::<math>\bigvee_{n\in\N} U_{n,a}=\top   \ \text{  for every }a\in\R</math>

:(where <math>\top</math> denotes the greatest element and <math>\bot</math> the smallest element of the frame.) The resulting locale is known as the "locale of surjective functions <math>\N\to\R</math>". The relations are designed to suggest the interpretation of <math>U_{n,a}</math> as the set of all those surjective functions <math>f:\N\to\R</math> with <math>f(n)=a</math>. Of course, there are no such surjective functions <math>\N\to\R</math>, and this is not a spatial locale.

== The theory of locales ==
We have seen that we have a functor <math>\Omega</math> from the [[Category of topological spaces|category of topological spaces and continuous maps]] to the category of locales. If we restrict this functor to the full subcategory of [[Sober space|sober spaces]], we obtain a [[full embedding]] of the category of sober spaces and continuous maps into the category of locales. In this sense, locales are generalizations of sober spaces.

It is possible to translate most concepts of [[point-set topology]] into the context of locales, and prove analogous theorems. Some important facts of classical topology depending on [[Axiom of choice|choice principles]] become choice-free (that is, [[Constructivism (mathematics)|constructive]], which is, in particular, appealing for computer science). Thus for instance, arbitrary products of [[Compact space|compact]] locales are compact constructively (this is [[Tychonoff's theorem]] in point-set topology), or completions of uniform locales are constructive. This can be useful if one works in a [[topos]] that does not have the axiom of choice.{{sfn|Johnstone|1983}} Other advantages include the much better behaviour of [[Paracompact space|paracompactness]], with arbitrary products of paracompact locales being paracompact, which is not true for paracompact spaces, or the fact that subgroups of localic groups are always closed.

Another point where topology and locale theory diverge strongly is the concepts of subspaces versus sublocales, and density: given any collection of dense sublocales of a locale <math>X</math>, their intersection is also dense in <math>X</math>.<ref>{{Cite book |last=Johnstone |first=Peter T. |title=Sketches of an Elephant |year=2002 |chapter=C1.2 Locales and Spaces}}</ref> This leads to [[John R. Isbell|Isbell]]'s density theorem: every locale has a smallest dense sublocale. These results have no equivalent in the realm of topological spaces.

==See also==
* [[Heyting algebra]]. Frames turn out to be the same as [[complete Heyting algebra|complete Heyting algebras]] (even though frame homomorphisms need not be Heyting algebra homomorphisms.)
* [[Complete Boolean algebra]]. Any complete Boolean algebra is a frame (it is a spatial frame if and only if it is atomic).
* Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the equivalence between [[sober space]]s and spatial locales, can be found in the article on [[Stone duality]].
* [[Whitehead's point-free geometry]].
*[[Mereotopology]].

==References==
{{reflist}}

===Bibliography===
A general introduction to pointless topology is
* {{Cite journal| issn = 0273-0979 | volume = 8| issue = 1| pages = 41–53| last = Johnstone| first = Peter T.| author-link= Peter Johnstone (mathematician)| title = The point of pointless topology| journal = Bulletin of the American Mathematical Society |series=New Series| access-date = 2016-05-09| year = 1983 | doi = 10.1090/S0273-0979-1983-15080-2| url = http://projecteuclid.org/euclid.bams/1183550014 | doi-access = free}}
This is, in its own words, to be read as a trailer for Johnstone's monograph and which can be used for basic reference:

* 1982: [[Peter Johnstone (mathematician)|Johnstone, Peter T.]] (1982) ''Stone Spaces'', Cambridge University Press, {{ISBN|978-0-521-33779-3}}.

There is a recent monograph

* 2012: Picado, Jorge, Pultr, Aleš [http://www.mat.uc.pt/%7Epicado/publicat/bookpage.html Frames and locales: Topology without points], Frontiers in Mathematics, vol. 28, Springer, Basel (extensive bibliography)

For relations with logic:

* 1996: [[Steve Vickers (computer scientist)|Vickers, Steven]], ''Topology via Logic'', Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.

For a more concise account see the respective chapters in:

* 2003: Pedicchio, Maria Cristina, Tholen, Walter (editors) Categorical Foundations - Special Topics in Order, Topology, Algebra and Sheaf Theory, ''Encyclopedia of Mathematics and its Applications'', Vol. 97, Cambridge University Press, pp.&nbsp;49–101.
* 2003: Hazewinkel, Michiel (editor) ''Handbook of Algebra'' Vol. 3, North-Holland, Amsterdam, pp.&nbsp;791–857.
* 2014: Grätzer, George, Wehrung, Friedrich (editors) ''Lattice Theory: Special Topics and Applications'' Vol. 1, Springer, Basel, pp.&nbsp;55–88.

[[Category:Category theory]]
[[Category:General topology]]