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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 38EQ

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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 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.

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