Arc-wise connectedness in semi-metric spaces
Robert Heath · Pacific Journal of Mathematics · 1962
A topological space S is said to be semi-metric if there is a distance function d for S with respect to which the topology of £ is invariant.A distance function d for S is a function from S X S to the nonnegative numbers such that, if each of x and y is a point of S, then (1) d(x, y) = 0 only in case x = y and (2) d(x, y) = d(y, x) [11; 18].The space is metric if the distance function also satisfies (3) d(x, y) -f d(y, x) ^ d(x, z) for each triple x, y, z of points of S. Note that every Moore space is regular and semi-metric.The set Uo(x) = {y> d(x, y) < c} is referred to herein as a c-neighborhood (with respect to d) of x.Cauchy complete is defined as in [11, p. 316].Topological space and regular are defined as in [9, pp.37 and 113].Terms not defined herein are used as in [14], [11], or [1].