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Definition 1. Let X and Y be two topological spaces. A map f: XY is proper if the preimage under f of any compact set is compact. Definition 2. A topological space X is said to be compactly generated if the following condition is satisfied: A subspace A is closed in X if and only if An K is closed in K for all compact subspaces KC X. Definition 3. A topological space X is locally compact if the following condition is satis- fied: For every point x E X, there is a compact subset KX that contains an (open) neighborhood of x. Let X X Y be the direct product of two topological spaces, and let p: X x Y→ X be the first-coordinate projection map. Show that p is proper if and only if Y is compact.

Definition 1. Let X and Y be two topological spaces. A map f: XY is proper if the preimage under f of any compact set is compact. Definition 2. A topological space X is said to be compactly generated if the following condition is satisfied: A subspace A is closed in X if and only if An K is closed in K for all compact subspaces KC X. Definition 3. A topological space X is locally compact if the following condition is satis- fied: For every point x E X, there is a compact subset KX that contains an (open) neighborhood of x. Let X X Y be the direct product of two topological spaces, and let p: X x Y→ X be the first-coordinate projection map. Show that p is proper if and only if Y is compact.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Definition 1.** Let \( X \) and \( Y \) be two topological spaces. A map \( f : X \rightarrow Y \) is proper if the preimage under \( f \) of any compact set is compact. **Definition 2.** A topological space \( X \) is said to be compactly generated if the following condition is satisfied: A subspace \( A \) is closed in \( X \) if and only if \( A \cap K \) is closed in \( K \) for all compact subspaces \( K \subseteq X \). **Definition 3.** A topological space \( X \) is locally compact if the following condition is satisfied: For every point \( x \in X \), there is a compact subset \( K \subseteq X \) that contains an (open) neighborhood of \( x \). Let \( X \times Y \) be the direct product of two topological spaces, and let \( p: X \times Y \rightarrow X \) be the first-coordinate projection map. Show that \( p \) is proper if and only if \( Y \) is compact.
Transcribed Image Text:**Definition 1.** Let \( X \) and \( Y \) be two topological spaces. A map \( f : X \rightarrow Y \) is proper if the preimage under \( f \) of any compact set is compact. **Definition 2.** A topological space \( X \) is said to be compactly generated if the following condition is satisfied: A subspace \( A \) is closed in \( X \) if and only if \( A \cap K \) is closed in \( K \) for all compact subspaces \( K \subseteq X \). **Definition 3.** A topological space \( X \) is locally compact if the following condition is satisfied: For every point \( x \in X \), there is a compact subset \( K \subseteq X \) that contains an (open) neighborhood of \( x \). Let \( X \times Y \) be the direct product of two topological spaces, and let \( p: X \times Y \rightarrow X \) be the first-coordinate projection map. Show that \( p \) is proper if and only if \( Y \) is compact.
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