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Definition. Let X and Y be topological spaces. A function or map f : X continuous function or continuous map if and only if for every open set U in Y, f-'(U) is open in X. → Y is a 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 E X and open set V containing f(x), there is an open set U containing x such that f(U) C V. Theorem 7.2. Let X,Y be topological spaces and yo E Y. The constant map f :X → Y defined by f(x) = yo is continuous. Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) — х is continиоus. Y be a continuous map between topological spaces, and let A be a subset of X. Then the restriction map f\a : A → Y defined by f\a(a) = f(a) is Theorem 7.4. Let f : X → continuous.

Definition. Let X and Y be topological spaces. A function or map f : X continuous function or continuous map if and only if for every open set U in Y, f-'(U) is open in X. → Y is a 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 E X and open set V containing f(x), there is an open set U containing x such that f(U) C V. Theorem 7.2. Let X,Y be topological spaces and yo E Y. The constant map f :X → Y defined by f(x) = yo is continuous. Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) — х is continиоus. Y be a continuous map between topological spaces, and let A be a subset of X. Then the restriction map f\a : A → Y defined by f\a(a) = f(a) is Theorem 7.4. Let f : X → continuous.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Could you explain how to show 7.4 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 the following are equivalent: Y be a function. Then (1) The function f is continuous. (2) For every closed set K in Y, the inverse image f-1(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 E X and open set V containing f(x), there is an open set U containing x such that f(U) V. Theorem 7.2. Let X,Y be topological spaces and yo E Y. The constant map f :X → Y defined by f(x) = yo is continuous. Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) — х is сontinuous. Theorem 7.4. Let f : X A be a subset of X. Then the restriction map f\a : A → Y defined by f\a(a) = f(a) is Y be a continuous map between topological spaces, and let сontinuous.
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 the following are equivalent: Y be a function. Then (1) The function f is continuous. (2) For every closed set K in Y, the inverse image f-1(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 E X and open set V containing f(x), there is an open set U containing x such that f(U) V. Theorem 7.2. Let X,Y be topological spaces and yo E Y. The constant map f :X → Y defined by f(x) = yo is continuous. Theorem 7.3. Let X c Y be topological spaces. The inclusion map i : X → Y defined by i(x) — х is сontinuous. Theorem 7.4. Let f : X A be a subset of X. Then the restriction map f\a : A → Y defined by f\a(a) = f(a) is Y be a continuous map between topological spaces, and let сontinuous.
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