Local connection in locally compact spaces

Abstract

It was proved by Hurewiczl that a compact space which is both LC1 and lc is LCD. In the present paper the corresponding result for locally compact spaces is proved, (a) for uniform local connection, and (b) for relative local connection.2 The 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 1cP and LCP conditions is used, namely (for l1C), the existence of a function t(5, x) such that, given a compact set F in the neighbourhood U(x, t(5, x)) of any point x, there is a compact subset F' of U(x, 5) such that every g-cycle in F bounds in F', for 0_ q ? p; and analogously for LCP. It is shown that these are equivalent to the ordinary 1cP and LCP properties in locally compact (metric) spaces.