构造性是locale理论与经典拓扑学相比最主要的特点，locale形式的Tychonoff乘积定理就且桓龅湫偷睦印＝酶衤邸OPOS理论和范畴论的工具，不用选择公理或完全构造性地建立在经典拓扑学中密切依赖于选择公理的相应结论。这些结构的建立一方面将改进和推广经典拓扑学中的已有结论，另一方面也为机器证明在拓扑学中的应用奠定理论基础。.

We obtained a characterization of S-compact locales,as applications,an internal characterization of S-cpmpact spaces is given.A new definition of convergence of filters on a locale is.given,which is equivalent to topological convergence on special locales and can be applied to any locales.Some characterizations of compactness and descriptions of Cauchy completeness are obtained.We proved that the inverse limits of compact spatial locales is in general not spatial and a sufficient condition for spatiality is given.As applications we obtained a sufficient condition for.spatiality of arbitrary product cales.This answers an open problem.posed by J.Isbell in 1972.Some necessary and sufficient conditions for the Scott topology on a complete lattice to be sober are obtained.The concept of homotopy of maps of locales is given and the.properties of the category of homotopy class of locales are investigated. We obtained a contravariant functor from the category of homotopy class of locales to the category of groups and showed that the fundamental groups is a homotopy invariant in the category of locales