Abstract
A regular (weakly) transitive translation function for a fibre bundle induces an H-space antihomomorphism from the loop space of the base into the group of the bundle. The question when such translation functions exist has had three answers to date. E. Brown [1, p. 226] states that every bundle with paracompact base admits weakly transitive translation functions. However, E. Fadell has pointed out an essential error in the proof. (Namely, [1, p. 227] v.(a), v.(f8) and v,,(a+o) are unrelated.) In Steenrod's book [4, p. 59] we find that if a bundle has totally disconnected group, then it admits transitive translation functions. J. Schlesinger [3] has proved a converse to this result for a restricted class of bundles. Schlesinger's theorem fails to contradict Brown's claim for the following reasons: (1) Brown employs Moore paths while Schlesinger uses ordinary (unit domain) paths. (2) Schlesinger discusses transitivity while Brown claims only weak transitivity. This paper removes the first of these distinctions; namely we prove:
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