Could you explain how to show 7.1 in detail? ### DefinitionLet \( X \) and \( Y \) be topological spaces. A function or map \( f : X \to Y \) 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 7.1Let \( X \) and \( Y \) be topological spaces, and let \( f : X \to Y \) 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 \( \overline{f(A)} \).4. For every \( x \in X \) and open set \( V \) containing \( f(x) \), there is an open set \( U \) containing \( x \) such that \( f(U) \subseteq V \).

Name: Could you explain how to show 7.1 in detail? ### DefinitionLet \( X \)
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Could you explain how to show 7.1 in detail? ### DefinitionLet \( X \) and \( Y \) be topological spaces. A function or map \( f : X \to Y \) is a **continuous function** or **continuous map** if and only if for every open set \( U \) in \( Y \), \(