Miscellaneous Facts about Open Functions and Continuous Functions

We adopt the following rules: n, m are elements of N, T is a non empty topological space, and M , M1, M2 are non empty metric spaces. The following propositions are true: (1) Let A, B, S, T be topological spaces, f be a function from A into S, and g be a function from B into T . Suppose that (i) the topological structure of A = the topological structure of B, (ii) the topological structure of S = the topological structure of T , (iii) f = g, and (iv) f is open. Then g is open. (2) Let P be a subset of Em T . Then P is open if and only if for every point p of Em T such that p ∈ P there exists a positive real number r such that Ball(p, r) ⊆ P.