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Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) = yo is continuous.

Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) = yo is 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 7.2 in detail?

**Definition.** Let \( X \) and \( Y \) be topological spaces. A function or map \( f: X \rightarrow Y \) is a continuous function or continuous map if and only if for every open set \( U \) in \( Y \), \( f^{-1}(U) \) is open in \( X \). **Theorem 7.1.** Let \( X \) and \( Y \) be topological spaces, and let \( f: X \rightarrow Y \) 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 \( f^{-1}(K) \) is closed in \( X \). 3. For every limit point \( p \) of a set \( A \) in \( X \), the image \( f(p) \) belongs to the closure of \( f(A) \). 4. For every \( x \in X \) and open set \( V \) containing \( f(x) \), there is an open set \( U \) containing \( x \) such that \( f(U) \subseteq V \). **Theorem 7.2.** Let \( X, Y \) be topological spaces and \( y_0 \in Y \). The constant map \( f: X \rightarrow Y \) defined by \( f(x) = y_0 \) is continuous.
Transcribed Image Text:**Definition.** Let \( X \) and \( Y \) be topological spaces. A function or map \( f: X \rightarrow Y \) is a continuous function or continuous map if and only if for every open set \( U \) in \( Y \), \( f^{-1}(U) \) is open in \( X \). **Theorem 7.1.** Let \( X \) and \( Y \) be topological spaces, and let \( f: X \rightarrow Y \) 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 \( f^{-1}(K) \) is closed in \( X \). 3. For every limit point \( p \) of a set \( A \) in \( X \), the image \( f(p) \) belongs to the closure of \( f(A) \). 4. For every \( x \in X \) and open set \( V \) containing \( f(x) \), there is an open set \( U \) containing \( x \) such that \( f(U) \subseteq V \). **Theorem 7.2.** Let \( X, Y \) be topological spaces and \( y_0 \in Y \). The constant map \( f: X \rightarrow Y \) defined by \( f(x) = y_0 \) is continuous.
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