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Exercise 3. Let X be a locally compact space. 1. Show that every closed subspace in X is locally compact. 2. Show that if X is Hausdorff then every open subspace in X is locally compact.

Exercise 3. Let X be a locally compact space. 1. Show that every closed subspace in X is locally compact. 2. Show that if X is Hausdorff then every open subspace in X is locally compact.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...
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3. Deduce. Exercise 3. Let X be a locally compact space. 1. Show that every closed subspace in X is locally compact. 2. Show that if X is Hausdorff then every open subspace in X is locally compact. 3. In usual R², say why {(0,0)} and {(x, y) = R²; x>0} are locally compact. 4. In the usual topology again, is {(0,0)} U {(x, y) = R²; x>0} locally compact? 5. Deduce.
Transcribed Image Text:3. Deduce. Exercise 3. Let X be a locally compact space. 1. Show that every closed subspace in X is locally compact. 2. Show that if X is Hausdorff then every open subspace in X is locally compact. 3. In usual R², say why {(0,0)} and {(x, y) = R²; x>0} are locally compact. 4. In the usual topology again, is {(0,0)} U {(x, y) = R²; x>0} locally compact? 5. Deduce.
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