2nd Edition

ISBN: 9780134689517

Author: Munkres, James R.

Publisher: Pearson,

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In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

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Chapter 4.34, Problem 7E

A space X is locally metrizable if each point x of X has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space X is metrizable if it is locally metrizable. [Hint: Show that X is a finite union of open subspaces, each of which has a countable basis.]

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Chapter 4.34, Problem 6E

Chapter 4.34, Problem 8E

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Chapter 4 Solutions

Topology

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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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A space X is locally metrizable if each point x of X has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space X is metrizable if it is locally metrizable. [ Hint : Show that X is a finite union of open subspaces, each of which has a countable basis.]

Solution Summary: The author explains that a compact Hausdorff space X is metrizable if it is locally.

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Chapter 4 Solutions