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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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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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