2nd Edition

ISBN: 9780134689517

Author: Munkres, James R.

Publisher: Pearson,

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Textbook Question

Chapter 4.36, Problem 1E

Prove that every manifold is regular and hence metrizable. Where do you use the Hausdorff condition?

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Chapter 4.35, Problem 9E

Chapter 4.36, Problem 2E

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Chapter 4 Solutions

Topology

Ch. 4.30 - Show that l and I02 are not metrizable.Ch. 4.30 - Which of our four countability axioms does S...Ch. 4.30 - Which of our four countability axioms does in the...Ch. 4.30 - Let A be a closed subspace of X. Show that if X is...Ch. 4.30 - Prob. 10E Ch. 4.30 - Let f:XY be continuous. Show that if X is...Ch. 4.30 - Let f:XY be continuous open map. Show that if X...Ch. 4.30 - Show that if X has a countable dense subset, every...Ch. 4.30 - Show that if X is Lindelof and Y is compact, then...Ch. 4.30 - Give I the uniform metric, where I=[0,1]. Let...

Ch. 4.30 - Prob. 16E Ch. 4.30 - Prob. 17E Ch. 4.30 - Prob. 18E Ch. 4.31 - Show that if X is regular, every pair of points of...Ch. 4.31 - Show that if X is normal, every pair of disjoint...Ch. 4.31 - Show that every order topology is regular.Ch. 4.31 - Prob. 4E Ch. 4.31 - Prob. 5E Ch. 4.32 - Which of the following spaces are completely...Ch. 4.32 - Prob. 8E Ch. 4.32 - Prove the following: Theorem: If J is uncountable,...Ch. 4.32 - Prob. 10E Ch. 4.33 - Examine the proof of the Urysohn lemma, and show...Ch. 4.33 - a Show that a connected normal space having more...Ch. 4.33 - Give a direct proof of the Urysohn lemma for a...Ch. 4.33 - Prob. 4E Ch. 4.33 - Prob. 5E Ch. 4.33 - Prob. 8E Ch. 4.34 - Give an example showing that a Hausdorff space...Ch. 4.34 - Give an example showing that a space can be...Ch. 4.34 - Let X be a compact Hausdorff space. Show that X is...Ch. 4.34 - Let X be a locally compact Hausdorff space. Is it...Ch. 4.34 - Let X be a locally compact Hausdorff space. Let Y...Ch. 4.34 - Check the details of the proof of Theorem 34.2.Ch. 4.34 - A space X is locally metrizable if each point x of...Ch. 4.34 - Show that a regular Lindelof space is metrizable...Ch. 4.35 - Show that the Tietze extension theorem implies the...Ch. 4.35 - In the proof of the Tietze theorem, how essential...Ch. 4.35 - Let X be metrizable. Show that the following are...Ch. 4.35 - Let Z be a topological space. If Y is a subspace...Ch. 4.35 - Prob. 5E Ch. 4.35 - Let Y be a normal space. The Y is said to be an...Ch. 4.35 - a Show the logarithmic spiral...Ch. 4.35 - Prove the following: Theorem. Let Y be a normal...Ch. 4.36 - Prove that every manifold is regular and hence...Ch. 4.36 - Let X be a compact Hausdorff space. Suppose that...Ch. 4.36 - Let X be a Hausdorff space such that each point of...Ch. 4.36 - Prob. 5E Ch. 4.SE - Consider the following properties a space may...Ch. 4.SE - Consider the following properties a space may...Ch. 4.SE - Prob. 3SE Ch. 4.SE - Consider the following properties a space may...

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Prove that every manifold is regular and hence metrizable. Where do you use the Hausdorff condition?

Solution Summary: The author explains that every manifold is regular and metrizable by the Urysohn Metrization Theorem.

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Chapter 4 Solutions