8. Let (X, d) be a metric space and let S ⊆ X. Prove that dS (x) = 0 if and only if x ∈ S

Question

8. Let (X, d) be a metric space and let S ⊆ X. Prove that dS (x) = 0 if and only if x ∈ S
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Solution 1

The statement is asking to prove that the distance from a point x to a set S in a metric space (X, d) is zero if and only if x is an element of S.

Here's a step-by-step proof:

  1. Assume that x is an element of S. By definition of the distance function dS(x), we have dS(x) = inf{d(x, y) : y ∈ S}. Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solv

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