Could you explain how to show 7.15 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 \( X \).**Definition.** Let \( f : X \rightarrow Y \) be a function between topological spaces \( X \) and \( Y \), and let \( x \in X \). Then \( f \) is **continuous at the point** \( x \) if and only if for every open set \( V \) containing \( f(x) \), there is an open set \( U \) containing \( x \) such that \( f(U) \subseteq V \). Thus a function \( f : X \rightarrow Y \) is continuous if and only if it is continuous at each point.**Theorem 7.15.** If \( X \) is compact and \( f : X \rightarrow Y \) is continuous and surjective, then \( Y \) is compact.**Theorem 7.18.** Let \( D \) be a dense set of a topological space \( X \), and let \( f : X \rightarrow Y \) be continuous and surjective. Then \( f(D) \) is dense in \( Y \).**Corollary 7.19.** Let \( X \) be a separable space, and let \( f : X \rightarrow Y \) be continuous and surjective. Then \( Y \) is separable.

Name: Could you explain how to show 7.15 in detail? **Definition.** Let \( X
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Could you explain how to show 7.15 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 \