Pointless topology
From Free net encyclopedia
In mathematics, pointless topology (also called point-free or pointfree topology) is an approach to topology which avoids the mentioning of points.
Contents |
General concepts
A traditional 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 {ai} of L has a supremum Vai such that
- <math>b \wedge \left( \bigvee a_i\right) = \bigvee \left(a_i \wedge b\right)</math>
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 doesn't 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.
See 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.
References
- P. T. Johnstone: The point of pointless topology. Bulletin American Mathematical Society, 8(1):41--53, 1983.es:Topología sin puntos