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Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Could you explain how to show 7.1 in detail?

Definition. Let X and Y be topological spaces. A function or map f : X → Y is a
continuous function or continuous map if and only if for every open set U in Y,
f-'(U) is open in X.
Theorem 7.1. Let X and Y be topological spaces, and let f : X → Y be a function. Then
the following are equivalent:
(1) The function f is continuous.
(2) For every closed set K in Y, the inverse image f-'(K) is closed in X.
(3) For every limit point p of a set A in X, the image f(p) belongs to f(A).
(4) For every x € X and open set V containing f(x), there is an open set U containing x
such that f(U) V.
Transcribed Image Text:Definition. Let X and Y be topological spaces. A function or map f : X → Y is a continuous function or continuous map if and only if for every open set U in Y, f-'(U) is open in X. Theorem 7.1. Let X and Y be topological spaces, and let f : X → Y be a function. Then the following are equivalent: (1) The function f is continuous. (2) For every closed set K in Y, the inverse image f-'(K) is closed in X. (3) For every limit point p of a set A in X, the image f(p) belongs to f(A). (4) For every x € X and open set V containing f(x), there is an open set U containing x such that f(U) V.
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