Question

Let (M, d) and (N, e) be metric spaces, and X ⊆ M. Let f: X → N. Then, f is a continuous map on the subspace (X, d) if and only if for every open set A in N, the inverse image f^(-1)(A) is open in X. Prove it using the topological definition of continuity

Answer 1

To prove that a function f: X → N is continuous if and only if the inverse image of every open set in N is open in X, we show that continuity implies the inverse image of open sets are open and that if the inverse image of every open set is open, the function must be continuous.

Step-by-step explanation:

To prove the statement regarding continuity of a function f: X → N using the topological definition of continuity, we need to show that a function is continuous if and only if the inverse image of every open set in N is open in X.

First, let's assume that f is continuous. By the topological definition of continuity, for any open set A in N, the inverse image f-1(A) must be open in X. This is because if f is continuous, and x is a point in f-1(A), there exists a δ such that the ball Bδ(x) is contained in f-1(A), making it an open set.

Conversely, if for every open set A in N, the inverse image f-1(A) is open in X, then for any point x in X and any neighborhood U of f(x) in N, there exists a neighborhood V of x in X such that f(V) ⊆ U. This confirms the continuity of f as every point x in X satisfies the condition of continuity.