Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only if it can be written as the union of a family of open balls of the form Br(x) = {y ∈ X : d(x, y) < r}.

Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only if it can be written as the union of a family of open balls of the form Br(x) = {y ∈ X : d(x, y) < r}.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 24CM

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Let (X, d) be a metric space and let A be a non-empty subset of X.

Prove that A is open if and only if it can be written as the union of a family of

open balls of the form Br(x) = {y ∈ X : d(x, y) < r}.

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