Page 77 - DMTH503_TOPOLOGY

Unit 7: Continuous Functions

7.1.4 Theorems and Solved Examples Notes

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Theorem 1: The function f: (X, J) (Y, U) is continuous iff f (V) is open in X for every open set
V in Y.
Proof: Let f: (X, J) (Y, U) be a map.
(i) Suppose f is continuous. Let G be an open subset of Y.

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To prove that f (G) is open in X.
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If f (G) = , then f (G) J.
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If f (G) , then x f (G) so that f(x) G.
Continuity of f f is continuous at x.
 H J s.t. x H and f(H) G.
x H f (G), H J.
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Thus we have shown that f (G) is a nhd of each of its points and so f (G) is J-open.
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Conversely, suppose that f : (X, J) (Y, U) is a map such that f (V) is open in X for each open set
V Y.
To prove that f is continuous.
Let V U be arbitrary.
Then, by assumption, f (V) is open in X.
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Take U = f (V), so that U J.
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i.e F(U) = f (f (V)) V, or f (U) V.
given any VU, U J s.t. f(U) V.
This proves that f is a continuous map.

Theorem 2: A map f : X  Y is continuous iff f (C) is closed in X for every closed set C  Y.
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A map f : (X, d)  (Y, p) be continuous iff f (F) is closed in X  F  Y is closed where (X, d) and
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(Y, p) are metric spaces.
Proof: Let f : X  Y be a continuous map.
To prove that f (c) is closed in X for each closed set C  Y.
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Let C Y be an arbitrary closed set.
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Continuity of f implies that f (Y  C) is open in X. (Refer theorem (1))
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i.e. f (Y)-f (C) is open in X.
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i.e. X  f (C) is open in X.
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or f (C) is closed in X.
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Conversely, suppose that f : (X, T)  (Y, U) is a map such that f (C) is closed for each closed set
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C  Y.
To prove that f is continuous.
Let G  Y be an arbitrary open set, then Y  G is closed in Y.

By hypothesis, f (Y  G) is closed in X.
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