On Local and Uniformly Local Topological Properties1

1. The main purpose of this paper is to show that certain properties of a metrizable space, such as local connectedness or local contractibility or LCn or local compactness, hold uniformly with a properly chosen metric.2 This result, while easy to prove, seems to be new, and often permits simplifications in proofs dealing with various kinds of local connectedness. In §2, we obtain an analogous (actually stronger) result dealing with coverings. Definition 1. An ordering3 a of the subsets of a space X is proper provided : (a) If WaV, then WQV; (b) If WCV, and VaR, then WaR; (c) If WaV, and VCR, then WaR. Definition 2. Let a be a proper ordering of the subsets of a metric space X. Then (a) X is locally of type a if, whenever xEX and Fis a neighborhood of x, then there exists a neighborhood W of x such that Wa V. (b) X is uniformly locally of type a if to every e>0 there corresponds a ô>0 such that S¡(x)aSt(x) for every xEX.* For example, if Wa V means that every pair of points in W is contained in a connected subset of V, then locally of type a means locally connected. If WaV means that W is contractible in V, then locally of type a means locally contractible. Other topological properties which arise in this manner are ra-LC (where WaV means that every continuous mapping of the «-sphere into W is homotopic to a constant mapping in V), LCn (i.e., ¿-LC for all k^n), and analogous kinds of homological connectedness.