Distances
and Metric Spaces| Distances
and Metric Spaces |
In General
* Idea: The most common way of mathematically realizing the intuitive
notion of closeness and distance.
$ Pseudodistance: A function d: X × X → R, X a
set, satisfying (1) d(p, q) 
0
and d(p, p)
= 0, positive semi-definiteness; (2) d(p, q)
= d(q, p),
symmetry; (3) d(p, q) 
d(p, r)
+ d(r, q), triangle inequality.
$ Pseudometric space:
A pair (X, d) with d a pseudodistance
on X.
* Remark: In such a "degenerate
metric space" two
points x 
y with d(x, y)
= 0 have to be equidistant from all other z's, because of the triangle
inequality, and thus "indistinguishable"; This is not
so in the Lorentzian case.
$ Distance: A positive-definite
pseudodistance d: X 
X → R.
$ Metric space: A pair
(X, d), with d a distance
on X.
* Relationships: It may
arise from a norm; In the topology induced by the distance, a metric space
is always paracompact.
@ References: Kurepa 63; Blumenthal 70; Schreider 74; Honig 95 [nonstandard].
> And other structures: see finsler geometry.
Related Notions and Results > s.a. cover; entropy;
types
of distances.
* Interesting maps: Isometries
and continuous maps (in the induced topology) are not so rich; Distance-decreasing
and 
-Lipschitz
maps are more
interesting.
* Baire category theorem:
A complete metric space is not the countable union of nowhere-dense sets; This
result can be stated as a theorem in Ramsey theory.
* Dilation of a map:
For f : X → Y, dil f :=
supx neq x' d(f(x), f(x'))
/ d(x,x');
dil f :=
limeps to 0 dil f |B(x,eps).
* Diameter: If A 
X is
a subset of a metric space, its diameter is diam(A):= l.u.b.{d(x, y)
| x, y in A}.
* Equicontinuity: A family 
of
functions on a metric space (X, d) is equicontinuous iff
for each 
> 0
there is a 
> 0
such that for all x and x' in X, and f in
, d(x, x') < implies
|f(x)–f(x')| < .
$ Outer measure of a set: The d-dimensional outer measure of A X is md(A):=
limeps to 0 inf
i (diam Si)d,
over all countable coverings of A by closed
spheres Si of
diameter < .
@ Other structure on metric spaces: Parthasarathy 67 [probability measures];
Penot JGP(07) [tangent vectors and differentials of mappings].
New Distances out of Old
* Sum and sup: If di are
distances on X (or even if all
but one of them are pseudodistances), then two new distances on X are
d(x, y):= i ai di(x, y),
with ai > 0
for all i , and d(x, y):=
supi di(x, y)
.
* On subsets A of X:
The induced distance dA(a, b):= dX(a, b)
is always available; In addition, if dX is
induced by a length structure,
we
can choose to first induce a length structure on the subset, d1,A(a, b):=
infgamma {l(
)
| a, b in im() A}.
Space of Metric Spaces > s.a. distance
between manifolds with metric.
@ Measure: Kondo DG&A(05)
Generalizations
* Lorentzian metric space:
A pair (M, d), with d: M M → R+ 
{
},
such that (i) d(x, x) = 0; (ii) d(x, y) > 0
implies d(y, x) = 0; (iii) d(x, y) d(y, z) > 0
implies that d(x, z) d(x, y)
+ d(y, z), the "reverse triangle inequality";
Examples of Lorentzian distance are the timelike geodesic distance between
two points, or the volume of their Alexandrov set.
* Quantum metric space: A C*-algebra (or more generally an order-unit
space) equipped with a generalization of the Lipschitz seminorm on functions
which is defined by an ordinary metric.
@ Lorentzian distance: Erkekoglu et al GRG(03)-
[level sets]; in Noldus
CQG(04)gq/03; > s.a.
causality conditions,
world function.
@ Quantum metric space: Rieffel MAMS(04)m.OA/00
[Gromov-Hausdorff distance].
@ Other: Mizokami & Suwada T&A(05)
[and their resolutions]; Kopperman et al T&A(09) [partial metric spaces, completion].
A qué le llaman distancia, eso me habrán de explicar – Atahualpa Yupanqui
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jul 2009