On Fibre Spaces. I

In subsequent papers I propose to investigate various properties of fibre spaces. The object of the fundamental Hurewicz-Steenrod definition is to state a minimum set of readily verifiable conditions under which the covering homotopy theorem holds. An apparent defect of their definition is that it is not topologically invariant. In fact, for topological space X and metrizable non-compact space B the property "X is a fibre space over B" depends on the metric of B. The object of this note is to give a topologically invariant definition of fibre space and to show that (when B is metrizable) X is a fibre space over B in this sense if and only if B has a metric in which X is a fibre space over B in the sense of Hurewicz-Steenrod. Since the definition of fibre space is controlled by the covering homotopy theorem, an essential part of my program is to give a topologically invariant definition of uniform homotopy. Let 7T be a continuous mapping of a topological space X into another topological space B. Let A=A(J3) denote the diagonal set 2^&£.B(&, b) of the product space BXB and let w denote the mapping of XXB into BXB which is induced by the mapping T according to the rule ïr(x, b) = {ir{x)t b). Thus the graph G of T is the set 7r (A), and 7T~(C7) is a neighborhood of G whenever U is a neighborhood of A. Any neighborhood U of A determines uniquely a covering of B by neighborhoods Nu(J>) according to the rule b'£:Nu(b) when (&, b')&J.