On Generalising the Notion of Fibre Spaces to Include the Fibre Bundles1

L Introduction. There are two different notions of fibering that have been investigated in the realm of topology. The one is the notion of a fibre bundle, defined by Whitney [ll]2 and improved by Steenrod [8]; the other is that of a fibre space, introduced by HurewiczSteenrod [6] and generalized by Fox [3]. These two notions do not coincide, as remarked by Steenrod, [8, p. 302]; there are fibre spaces that are not fibre bundles, while it is still unsolved whether every fibre bundle is a fibre space. According to Fox [3, p. 555], the object of introducing the definition of fibre spaces is to state a minimum set of readily verifiable conditions under which the covering homotopy theorem holds. It seems to the author that this set of conditions, given by Hurewicz-Steenrod [6] and Fox [3], is not minimum in the sense that it does not naturally apply to the fibre bundles for which the covering homotopy theorem was proved by Steenrod [8, p. 303] in a somewhat weaker form. This weak theorem of covering homotopy of Steenrod works satisfactorily in nearly all the applications where the base space is normal Hausdorff. In order that the weak theorem of covering homotopy hold, the existence of a unified slicing function in the definition of HurewiczSteenrod [ö] and in that of Fox [3] is unnecessarily strong. The object of the present paper is to give a generalization of the notion of a fibre space by localizing the slicing function. It will be proved that the generalized class of fibre spaces will include all fibre bundles and that the weak theorem of covering homotopy will still hold. As mentioned above, there are two forms of the covering homotopy theorem. In the "weak" form, the set being deformed is compact and the homotopy is unrestricted. In the "strong" form, the set is unrestricted but the homotopy is required to be uniform. Since any homotopy of a compact space is uniform, the "strong" theorem implies the "weak" theorem. If the set is completely regular, any uniform homotopy into a compact Hausdorff space can be extended to