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#deck:Math::Metric spaces::Open balls
#tags:math metric-spaces open-balls
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What is the open ball B(p, r)?	The set of points at distance less than r from p. Strict: the rim is not in it.	what
When is a set open?	When every point of it keeps some open ball around it inside the set.	what
Why is an open ball an open set?	For x in B(p, r), the ball B(x, r - d(p, x)) fits inside, by the triangle inequality.	why
Is the closed disc x^2 + y^2 <= 9 open?	No: every ball around (3, 0) contains (3 + r/2, 0), which is outside. One rim point is enough.	trap
State the chain between the plane's three metrics.	cheb <= eucl <= taxi <= 2 cheb, on every pair of points; three one-line inequalities with a = |dx| and b = |dy|.	what
How do the balls of the three metrics nest?	B_taxi(p, r) inside B_eucl(p, r) inside B_cheb(p, r) inside B_taxi(p, 2r).	what
What are equivalent metrics, and how are they usually proved equivalent?	Metrics with the same open sets. A two-sided bound c d1 <= d2 <= C d1 nests each ball of one kind inside a ball of the other.	why
Which pair of points makes taxi = 2 cheb?	Any pair with |dx| = |dy|, such as (0, 0) and (1, 1): taxicab 2, Chebyshev 1. The factor 2 is sharp.	problems
What are the balls of the discrete metric?	B(p, r) is {p} for r <= 1 and the whole space for r > 1. So every subset is open.	what
Is the discrete metric equivalent to the Euclidean one?	No. {p} is open under the discrete metric and not under the Euclidean one, where every ball of positive radius contains p + (r/2, 0).	trap