Equicontinuity and $n$-length

Abstract

Let (M, p) be a pseudo-metric space. We shall obtain a necessary and sufficient condition that a collection of curves can be parametrized in such a manner that the collection of parametrizations be equicontinuous. This result can be extended to the case where p is a quasi-pseudo-metric. The ,-length defined here differs inessentially from that originally defined by M. Morse in [M]. We also use ideas of [F] and [S]. Let I= [a, b]. If uCI, then let I.= [a, u]. If (N, a) is a pseudometric space and iff: I-+N, then let w(f; J) = sup { r(f(u) ,f(v)) u, v CJ whenever J is an interval contained in I. Let C(I) be the space of continuous functions on I into (AIl, p) metrized by ar where or(x, y) =sup{p(x(u), y(u)) I uC I}. For each positive integer n and x C C(I), let