Definition 1. Let X and Y be two topological spaces. A map f: X → Y is proper if thepreimage under f of any compact set is compact.Definition 2. A topological space X is said to be compactly generated if the followingcondition is satisfied: A subspace A is closed in X if and only if An K is closed in K for allcompact 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 KCX that contains an (open)neighborhood of x.Let f: X Y be a proper, continuous map, and assume Y is a compactly generated,Hausdorff space. Show that f is a closed map.

Proper: Let X and Y be two topological spaces. A map f:X→Y is proper if the preimage under f of any…

Answered: Let f: X Y be a proper, continuous map,…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

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Definition 1. Let X and Y be two topological spaces. A map f: X 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 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 KC X 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: X → 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 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 KC X that contains an (open) neighborhood of x. Suppose X is compact, and Y is Hausdorff. Show that every continuous map f: X → Y is both closed and proper.
Definition 1. Let X and Y be two topological spaces. A map f: X→ 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 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 KCX that contains an (open) neighborhood of x. Let f: X → Y be a continuous map from a compact space X to a Hausdorff space Y. Let C be a closed subspace of Y, and let U be an open neighborhood of f-¹(C) in X. Show that there is an open neighborhood V of C in Y such that f-¹(V) is contained in U.
Definition 1. Let X and Y be two topological spaces. A map f: X→ 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 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 E X, there is a compact subset KCX that contains an (open) neighborhood of x. Let (X, d) be a metric space, and let f: X X be a continuous function which has no fixed points. (a) If X is compact, show that there is a real number >0 such that d(x, f(x)) > €, for all r € X. (b) Show that the conclusion in (a) is false if X is not assumed to be compact.
1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show that X is not compact.
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Let f: X Y be a proper, continuous map, and assume Y is a compactly generated, Hausdorff space. Show that f is a closed map.