Pointless topology


Pointless topology

In mathematics, pointless topology (also called point-free or pointfree topology) is an approach to topology which avoids the mentioning of points.

General concepts

Traditionally, a topological space consists of a set of points, together with a system of open sets. These open sets with the operations of intersection and union form a lattice with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.

Categories of frames and locales

Formally, a frame is defined to be a lattice "L" in which finite meets distribute over arbitrary joins, i.e. every (even infinite) subset {"a"i} of "L" has a supremum ⋁"a""i" such that

:b wedge left( igvee a_i ight) = igvee left(a_i wedge b ight)

for all "b" in "L". These frames, together with lattice homomorphisms which respect arbitrary suprema, form a category. The dual of the category of frames is called the category of locales and generalizes the category Top of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every continuous map between topological spaces "X" and "Y" induces a map between the lattices of open sets "in the opposite direction" as for every continuous function "f": "X" → "Y" and every open set "O" in "Y" the inverse image "f" -1("O") is an open set in "X".

Relation to point-set topology

It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems.While many important theorems in point-set topology require the axiom of choice, this is not true for their analogues in locale theory.This can be useful if one works in a topos which does not have the axiom of choice.

The concept of "product of locales" diverges slightly from the concept of "product of topological spaces", and this divergence has been called a disadvantage of the locale approach.Others claim that the locale product is more natural and point to several of its "desirable" properties which are not shared by products of topological spaces.

For almost all spaces (more precisely for sober spaces) the topological product and the localic product have the same set of points. The products differ in how equality between sets of open rectangles (=the canonical base for the product topology) is defined: equality for the topological product means the same set of points is covered;equality for the localic product means provable equality using the frame axioms. As a result two open sublocales of a localic product may contain exactly the same points without being equal.

A point where locale theory and topology diverge much more strongly is the concept of subspaces vs. sublocales.The rational numbers have "c" subspaces but 2"c" sublocales. The proof for the latter statement is due to John Isbell and uses the fact that the rational numbers have "c" many pairwise almost disjoint (= finite intersection) closed subspaces.

ee also

* Heyting algebra. A locale is a complete Heyting algebra.
* Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober spaces and spatial locales, are to be found in the article on Stone duality.
* Point-free geometry
* Mereology
* Tacit programming

References

*Johnstone, Peter T., 1983, " [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014 The point of pointless topology,] " "Bulletin of the American Mathematical Society 8(1)": 41-53.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Topology — (Greek topos , place, and logos , study ) is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with… …   Wikipedia

  • List of topology topics — This is a list of topology topics, by Wikipedia page. See also: topology glossary List of general topology topics List of geometric topology topics List of algebraic topology topics List of topological invariants (topological properties)… …   Wikipedia

  • General topology — In mathematics, general topology or point set topology is the branch of topology which studies properties of topological spaces and structures defined on them. It is distinct from other branches of topology in that the topological spaces may be… …   Wikipedia

  • Stone duality — In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they… …   Wikipedia

  • Mereotopology — In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries… …   Wikipedia

  • Order theory — For a topical guide to this subject, see Outline of order theory. Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as… …   Wikipedia

  • Tychonoff's theorem — For other theorems named after Tychonoff, see Tychonoff s theorem (disambiguation). In mathematics, Tychonoff s theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey… …   Wikipedia

  • Heyting algebra — In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras, named after Arend Heyting. Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded… …   Wikipedia

  • List of order theory topics — Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is less than or precedes another. An alphabetical list of many… …   Wikipedia

  • List of order topics — This is a list of order topics, by Wikipedia page.An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value, optimization (mathematics), domain theory.Basic… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.