Local connection in locally compact spaces

Abstract

It was proved by Hurewicz1 that a compact space which is both LC1 and lc" is LCn.In the present paper the corresponding result for locally compact spaces is proved, (a) for uniform local connection, and (b) for relative local connection.2The extension of Hurewicz's theorem to locally compact spaces is included in (b).The main difficulty in extending Hurewicz's methods is that his "Satz 6," on the passage from e-homotopy to true homotopy, cannot be carried over to locally compact spaces without substantial modification, even when uniform local connection is assumed.To overcome this a stronger form of the lcp and LO conditions is used, namely (for lcp), the existence of a function f (5, x) such that, given a compact set F in the neighbourhood U(x, f (5, x)) of any point x, there is a compact subset F' of U(x, 8) such that every g-cycle in F bounds in F', for Ogqgp;and analogously for LO.It is shown that these are equivalent to the ordinary lcp and LO properties in locally compact (metric) spaces.1. Definitions.It is assumed once for all that the containing space X is locally compact, and has metric p.* Homologies are relative to integral coefficients; cycles in X are Vietoris-cycles (but finite cycles are chains on some simplicial complex with vertices in X).The statement that T bounds in E means that T bounds in a compact subset of E. p denotes an integer not less than 0.The letter F, with various suffixes, always denotes a compact set.If G is open, the statement "FQG with a margin a" means that a>0, and Cl(f/(P, a)) is compact and contained in G* The existence of margins for every such F and G is ensured by the local compactness of X.A set Ex is acp rel.E2 ("acyclic up to p rel. E2") if every g-cycle in Ei bounds in E2, for Ogqgp.Ei is as" rel.E2 ("aspherical up to p rel. E2") if every mapping of the g-sphere S" into Ei is null-homotopic in E2, for Ogqgp.The set Ei is strongly acp (or strongly asp) rel.E2 if, given any F in Ei there is an F' in E2 such that F is acp (or asp) rel.F'.Presented to the Society, July 19, 1948; received by the editors October 25, 1948. 1 Hurewicz [4], denoted hereafter by H. Numbers in brackets refer to the bibliography at the end of the paper.* Problem 4 of Eilenberg and Wilder [3] is thereby settled affirmatively.' Use is made at one point (Theorem 1, (B)) of a local separability condition.4 Cl.(X) denotes the closure of X.