Topological Spaces-Continuous Functions between Topological Spaces
المؤلف:
David R. Wilkins
المصدر:
Algebraic Topology
الجزء والصفحة:
6
6-7-2017
1932
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).
The result therefore follows immediately from the definitions of continuity and closed sets.
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