Re: Contractible metric space

From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 02/25/05 

Date: 25 Feb 2005 22:49:55 GMT

Julien Santini wrote:
> What is meant by "unique"?

There is only one object satisfying the properties. Formally, if we
let P(x) be a predicate corresponding to the properties, then
 P(A) and P(B) => A = B.

> What is meant by arc ? (traditionally an arc is a continuous map
> defined on [0,1] right ?)

Almost: that's a path. An arc is more restricted: its function must
be bijective and continuous in both directions. In mathematical
formalism:

f:[0,1] -> S is an arc in M iff S is a subspace of M and f is
bijective, continuous, and has continuous inverse.

> And what about "congruent [...] to a line segment ?"

There exists an distance preserving bijection between an interval of R
(i.e. a line segment) and the arc. Formally:

A subspace S of M is congruent to a line segment iff there exists a,b
in R and f:[a,b] -> S such that f is bijective and for all x,y in [a,b],
d(f(x),f(y)) = |x-y|.

- Tim

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