A metric space (X, d) is called separable if it contains a countable dense subset, that is, if there exists a countable subset ECX such that E = X. Prove that every compact metric space is separable. Hint: For each n € N, consider an open cover consisting of neighborboods of radius ¹.
A metric space (X, d) is called separable if it contains a countable dense subset, that is, if there exists a countable subset ECX such that E = X. Prove that every compact metric space is separable. Hint: For each n € N, consider an open cover consisting of neighborboods of radius ¹.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:A metric space (X, d) is called separable if it contains a countable dense subset,
that is, if there exists a countable subset ECX such that E = X. Prove that
every compact metric space is separable.
Hint: For each nN, consider an open cover consisting of neighborboods of
radius 12.
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