Formally, we define a frame to be a lattice which has a largest and smallest element and in which every (even infinite) subset has a supremum. These frames, together with lattice homomorphisms which respect suprema of infinite sets, form a category; the opposite category of this 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. |
Formally, we define a frame to be a lattice L in which every (even infinite) subset {ai} has a supremum Vai such that :b ^ (V ai) = V (ai ^ b) 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 analoga 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. |
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Formally, we define a frame to be a lattice L in which every (even infinite) subset {ai} has a supremum Vai such that
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 analoga 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.