The metric space M is separable if it contains a countable dense subset. [Note the confusion...

The metric space M is separable if it contains a countable dense subset. [Note
the confusion of language: “Separable” has nothing to do with “separation.”]
(a) Prove that R^m is separable.
(b) Prove that every compact metric space is separable.

Expert Solution

Colby Messinger answered 2 years ago

Show that id D is dense metric space of X and if all cauchy sequences Yn...

Show that id D is dense metric space of X and if all cauchy sequences Yn from a sense set D converge in D then X is complete.

Suppose K is a nonempty compact subset of a metric space X and x ∈ X....

Suppose K is a nonempty compact subset of a metric space X and x ∈ X. (i) Give an example of an x ∈ X for which there exists distinct points p, r ∈ K such that, for all q ∈ K, d(p, x) = d(r, x) ≤ d(q, x). (ii) Show, there is a point p ∈ K such that, for all other q ∈ K, d(p, x) ≤ d(q, x). [Suggestion: As a start, let S = {d(x,...

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The metric space M is separable if it contains a countable dense subset. [Note the confusion...

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