Definition: A function f: X → Y from a topological space X to a topological space Y is said to be continuous if  f⁻¹ (V ) is an open set in X for every open set V in Y , where

                                        f⁻¹ (V ) ≡ {x ∈ X : f(x) ∈ V }.

A continuous function from X to Y is often referred to as a map from X to Y .

Lemma 1.5 Let X, Y and Z be topological spaces, and let f: X → Y and g: Y → Z be continuous functions. Then the composition g ◦ f: X → Z of the functions f and g is continuous.

Proof Let V be an open set in Z. Then g⁻¹ (V ) is open in Y (since g iscontinuous), and hence f⁻¹ (g⁻¹ (V )) is open in X (since f is continuous).

But f⁻¹ (g⁻¹ (V )) = (g ◦ f) ⁻¹

(V ). Thus the composition function g ◦ f iscontinuous.

Lemma 1.6 Let X and Y be topological spaces, and let f: X → Y be afunction from X to Y . The function f is continuous if and only if f⁻¹ (G) is closed in X for every closed subset G of Y .

Proof If G is any subset of Y then

                 X f⁻¹ (G) = f⁻¹ (Y G)

(i.e., the complement of the preimage of G is the preimage of the complement of G).