(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

To prove that every metric space is regular. :)

3. The attempt at a solution

So, a regular space satisfies the T1 and T3-axioms.

For T1: Let a, b be two distinct points of a metric space (X, d). Then d(a, b) > 0, and let r = d(a, b)/2. Then the open ball K(a, r) is a neighborhood of a which doesn't contain b, and K(b, r) is a neighborhood of b which doesn't contain a.

For T3: Let A be a closed subset of X, and let b be in X\A. Since we proved that T1 holds, for every x in A and for b there exists a neighborhood of Ux which doesn't contain b. A is a subset of the union U of these neighborhoods (U is a neighborhood of A). Now, define r = min{d(x, b)/2: x is in A}. Then K(b, r) and U are disjoint.

I hope this works.

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# Homework Help: Every metric space is regular

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