Open subset inclusion graph of a topological space

In this paper we introduce a graph topological structure, called open subset inclusion graph of a topological space J(tau) on a finite set X, where the vertex set is the collection of nonempty proper open subsets of a topological space and two vertices U-1, U-2 are adjacent or U-1 similar to U-2 or (U-1, U-2) is an element of E, if either U-1 subset of U-2 or U-2 subset of U-1. The diameter, girth,connectivity, maximal independent sets, different variants of domination number, clique number and chromatic number, degree of J(tau), edge and vertex connectivity of J(tau) are studied