ized p-sphere with a fixed pole yo. Following G. W. Whitehead (10), we shall denote by GP(X) the mapping space XSP, which is the totality

of (continuous) maps of SP into X endowed with compact-open topology. Let ir: GP(X)-*X be defined by ir(f) =f(yo), (f EGP(X)), and let FP(X, x) =r-1(x) for each xEX. Consider now the mapping space B(X) consisting of all the maps of yo into X. There is a natural map P: GP(X) ->B(X) defined by p(f) =f I yo for every fEGP(X). It is well known (cf. [3, pp. 83-84]) that p has the path lifting property. Clearly, the space X can be identified with B(X) in a natural way. The map r is then identified with p. Consequently ir: GP(X)-+X is a fibre map of GP(X) onto X having the absolute covering homotopy property [3, p. 82]. For each xEX, the fibre in GP(X) over x is FP(X, x). The arc components of FP(X, x) are elements of the pth homotopy group '7rp(X, x) of X at x. Denote by GP(X) the arc component of GP(X) which contains a==FP(X, x)e'zrp(X) (cf. [10]). If X is arcwise connected, then GP(X) is also a fibre space over X. The restriction 7ra =ir I GP(X) is a fibre map of GP(X) onto X. The homo