The Cantor Set 1

The articles [10], [4], [12], [11], [7], [13], [2], [3], [6], [8], [5], [1], and [9] provide the notation and terminology for this paper. Let Y be a set and let x be a non empty set. Note that Y 7−→ x is non-empty. Let X be a set and let A be a family of subsets of X . The functor UniCl(A) yields a family of subsets of X and is defined as follows: (Def. 1) For every subset x of X holds x ∈ UniCl(A) iff there exists a family Y of subsets of X such that Y ⊆ A and x = ⋃ Y. Let X be a topological structure. A family of subsets of X is said to be a basis of X if: (Def. 2) It⊆ the topology of X and the topology of X ⊆ UniCl(it). One can prove the following propositions: