Difference between revisions of "Open ball"

Revision as of 20:27, 12 February 2015 (view source) Alec (Talk | contribs) (Created page with "{{Definition|Topology|Metric Space}} For a metric space <math>(X,d)</math> an "open ball" of radius <math>r</math> centred at <math>a</math> is the set <math...") Newer edit →

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Revision as of 20:27, 12 February 2015

For a metric space

$$(X,d)$$ an "open ball" of radius $$r$$ centred at $$a$$ is the set $$\{x\in X|d(a,x)\lt r\}$$ , it can be denoted several ways. I frequently encounter

$$B_r(a)=B(a;r)=\{x\in X|d(a,x)\lt r\}$$ and use $$B_r(a)$$

Proof that an open ball is open

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TODO: The proof is really easy, just show a smaller ball fits inside, thus the open ball is a neighborhood to all of its points

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