Pointless topology
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Pointless topology (also called point-free or pointfree topology) is an approach to topology which avoids the mentioning of points. A traditional topological space consists of a set of "points", together with a system of "open sets". These open sets 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. Some proponents claim that this new category has certain natural properties which make it preferable. 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.
Formally, we define a frame to be a lattice L in which every (even infinite) subset {ai} 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 and all sets {ai} of L. These frames, together with lattice homomorphisms which respect arbitrary suprema, form a category; the opposite category of the category of frames is called the category of locales and generalizes the category of topological spaces. The reason that we take the opposite category is that every continuous map f : X → Y between topological spaces induces a map between the lattices of open sets in the opposite direction: every open set O in Y is mapped to the open set f -1(O) in X.
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.
References
- P. T. Johnstone: The point of pointless topology. Bulletin American Mathematical Society, 8(1):41--53, 1983.es:Topología sin puntos