J.
Heitzig
Between
metric spaces, the most powerful kinds of mappings are certainly Lipschitz
continuous maps and isometric embeddings. Both notions cannot be expressed in
terms of, say, quasi-uniformities, due to their lack of an additive structure.
This
talk is about several properties of maps between distance spaces more general
than metric spaces, whose distance functions are allowed to take values in
quasi-ordered monoids other than the real line.
It
is argued that some of these properties are quite close to Lipschitz continuity
and isometry, and may serve as their substitutes in quasi-uniform spaces,
because each quasi-uniformity comes from a multi-qp-metric (i.e., a
distance function taking values in a power of the real line). In addition to
many examples, a topological and a geometrical theorem will be given in
particular:
(1)
Call a map h
between qp-metric spaces (X,
d) and (Y, e)
strongly uniformly continuous iff for all
e
> 0, there is
d
> 0
such that whenever
d(x1, y1) + . . .
+ d(xn, yn)
£
d,
then e(hx1,
hy1) + . . . + e(hxn,
hyn) £
e.
Theorem.
If X
is a subset of a Banach space whose closure is convex, strong uniform
continuity coincides with Lipschitz continuity for all maps from
X into qp-metric
spaces.
(2)
Theorem. The continuous maps h
from Rn
(n > 1) to Rm
that fulfill the implication
if | x - y | = | z - w | then | hx -
hy | = | hz - hw |
are
exactly the similarity maps.