- METRIC SPACES AND LENGTH SPACES 389
REMARK G.1 (Scaling metric spaces). If (X, d) is a metric space, then
so is (X, ad), where a > 0 and (ad) (x, y) ~ a· d (x, y). Two important
aspects of metrics are the behaviors of ad as a -+ 0 (where large scalesbecome 'visible') and as a-+ oo (where small scales become visible).
More generally, we say that a pair ( X, d), where Xis a set and where
d: X x X-+ JR U { oo} is a map, is a quasi-metric space^1 if the triangle
inequality holds, i.e., for every x, y, z EX(G.l) d(x,z) ~ d(x,y) +d(y,z).
In particular, d may be neither nonnegative nor symmetric and d may be
infinite.
REMARK G.2. Let ( X, T, J) be a triple where Xis a set, where T C
X x X is a transitive relation on X,^2 and where d : T -+ JR is a (finite)
function such that for every (x, y), (y, z) ET
(G.2) d(x,z) ~ d(x,y) +d(y,z)
(by transitivity, (x, z) ET). If we extend d to a map from Xx X to JRU{oo}
by defining d = oo on (Xx X) -T, then the triangle inequality still holds.
That is, ( X, J) is a quasi-metric space.
If for a quasi-metric space ( X, J), we also have
(1) (distance is nonnegative and finite) 0 ~ d < oo,
( 2) (distance from a point to itself is zero) d ( x, x) = 0 for all x E X,
and
(3) (distance is symmetric) d (x, y) = d (y, x) for all x, y EX,then d is called a pseudo-metric^3 and ( X, J) is called a pseudo-metric
space.
In general, given a pseudo-metric space ( X, J), we may define the equiv-
alence relation rv on X by x rv y if and only if d (x, y) = 0. Let X' ~ X/ rv
be the quotient space. Define d' : X' x X' -+ [O, oo) by
(G.3) d' ([x], [y]) ~ d (x, y).
Then d' is a well-defined metric on X'.^4 We call ( X', d') the (quotient)
metric space induced by the pseudo-metric space ( X, J).
(^1) A more standard use of the term 'quasi-metric space' is for a metric space without
the axiom of symmetry.
(^2) A relation Ron Xis a subset of Xx X. It is transitive if (x, y), (y, z) ER implies
(x, z) ER.
(^3) We also use the terminology of quasi-distance and pseudo-distance.
(^4) See Proposition 1.1.5 on p. 2 of [18]; there the proof is left as an exercise.