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## Homework Statement

Let A be a finite open subset of a metric space M. Prove that every point of A is an isolated point of M.

## Homework Equations

A point x is isolated if there exists some r such that the open of radius r centered at x consists of x alone.

alternative definition of isolated:the set {x} is open

## The Attempt at a Solution

Let A = {a1, a2, ... an}. Let r = min{D(a

_{i}, a

_{j}): for all i≠j, 1≤i<j≤n}. Then for all x in A, S

_{r}(x)INTERSECT M = {x}, so every point is an isolated point of M.

So I think my teacher ok'd this, but now that I think about it, I have a question. What does it mean to be isolated in M? I get that every point in A is isolated, but why is it isolated in M?

Also, the first definition of isolated makes intuitive sense to me (the idea of isolated in english seems to fit this definition). The second definition does not make intuitive sense to me. Does anyone have an intuitive way to interpret the second definition?

This subject (metric spaces) has been a little difficult for me :(