Despite the fact noted above that in general a uniformly continuous map need
not be uniformly homologous, there is a large class of spaces, which includes all
complete connected Riemannian manifolds, in which this implication does hold.
These are the path metric spaces (espaces de longueur) of Gromov , in which
the distance between two points is defined as the infimum of the lengths of curves
(2.2) P R O P O S I T I O N : Let M be a path metric space, N a metric space, and
f:M-+N a uniformly continuous map; then f is uniformly bornologous.
P R O O F : Since / is uniformly continuous, there is 6 0 such that if x,x' £
M, d(x,xf) 8 then d(f(x), f(x')) 1. Now by definition of a path metric
space, if x,x' are points of M with d(x,x') R, then there is a continuous curve
7 in M with endpoints x and x' and length less than R. Subdividing 7, we find
that there is a chain x = xo,x\,... , xn = x' of points of M with d(xj, Xj+\) 6
and n R/6 -f 1. Hence, by the triangle law,
d(f(z),f(z')) J2d(f(xj)J(xH1)) n j + 1
so f is uniformly bornologous. •
In fact this argument shows that / is eventually Lipschitz.
(2.3) DEFINITION: A metric space M is a proper metric space if closed bounded
subsets of M are compact.
Such a space is necessarily complete.
(2.4) DEFINITION: The category UBB ('uniformly bornologous Borel') is the
category whose objects are proper metric spaces and whose morphisms are uni-
formly bornologous Borel maps which are proper in the sense that the inverse
image of a relatively compact set is relatively compact.
Most of our theory will be functorial on the category UBB. We will adopt
the convention that the word 'space' will always (unless otherwise specified)
mean 'object in the category UBB' and the word 'morphism' will always (unless
otherwise specified) mean 'morphism in the category UBB'.
For some purposes we will need a more restricted class of morphisms. The
category UBC ('uniformly bornologous continuous') is defined to be the sub-
category of UBB with the same objects but with uniformly bornologous proper
continuous maps as morphisms. Again, we will abbreviate 'morphism in the
category U B C to 'continuous morphism'.
(2.5) DEFINITION: Let M and N be spaces, and let f,g: M —• N be morphisms.
We say that f and g are bornotopic, and write f ~ g, if there is a constant
R 0 such that for all x G M, d(f(x),g(x)) R. Further, if f: M - N
is a morphism, and there exists a morphism h: A' -+ M such that f o h ~
ITVJ ho f ~ 1M, we say that f is a bornotopy-equivalence, and that M and N