Components and Unions of Components

In this paper G1 will denote a non empty topological space and V , A will denote subsets of the carrier of G1. Let G1 be a non empty topological structure and let V be a subset of the carrier of G1. The functor Component(V ) yields a subset of the carrier of G1 and is defined by the condition (Def. 1). (Def. 1) There exists a family F of subsets of G1 such that for every subset A of the carrier of G1 holds A ∈ F iff A is connected and V ⊆ A and ⋃ F = Component(V ). One can prove the following propositions: (1) If there exists A such that A is connected and V ⊆ A, then V ⊆ Component(V ). (2) If it is not true that there exists A such that A is connected and V ⊆ A, then Component(V ) = ∅. (3) Component(∅(G1)) = the carrier of G1.