Baire Spaces , Sober Spaces 1 Andrzej

In the article concepts and facts necessary to continue formalization of theory of continuous lattices according to [11] are introduced. and [4] provide the notation and terminology for this paper. 1. PRELIMINARIES The following two propositions are true: (1) For all sets X, A, B such that A ∈ Fin X and B ⊆ A holds B ∈ Fin X. (2) For every set X and for every family F of subsets of X such that F ⊆ Fin X holds F ∈ Fin X. Let X be a non empty set. Let us observe that X is trivial if and only if: (Def. 1) For all elements x, y of X holds x = y.