Equicontinuity and $n$-length
Equicontinuity and $n$-length
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等连续性和$n$-长度
DOI：
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发表时间：
1969
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通讯作者：
E. Silverman
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作者：
E. Silverman
设（M, p）为伪度量空间。我们将得到曲线集合可以参数化的一个充分必要条件，使参数化集合是等连续的。这个结果可以推广到p是拟伪度量的情况。这里定义的长度与M. Morse在[M]中最初定义的长度没有本质上的不同。我们也使用[F]和[S]的概念。设I= [a, b]。如果uCI，则让i = [a, u]。如果（N, a）是伪度量空间，且iff: I-+N，则设w(f； J) =sup{r(f(u),f(v)) u, v CJ，当J是包含在I中的区间时，设C(I)是I上的连续函数的空间，该空间由ar度量，其中(x, y) =sup{p(x(u), y(u)) I uC I}。对于每一个正整数n和x C C(I)，令
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