On the Baire Category Theorem

Let T be a topological structure and let A be a subset of the carrier of T . Then IntA is a subset of T . Let T be a topological structure and let P be a subset of the carrier of T . Let us observe that P is closed if and only if: (Def. 1) −P is open. Let T be a non empty topological space and let F be a family of subsets of T . We say that F is dense if and only if: (Def. 2) For every subset X of T such that X ∈ F holds X is dense. The following proposition is true (1) Let L be a non empty 1-sorted structure, A be a subset of L, and x be an element of L. Then x ∈ −A if and only if x / ∈ A. Let us observe that there exists a 1-sorted structure which is empty. Let S be an empty 1-sorted structure. Note that the carrier of S is empty.