The theory of covering groupoids plays an important role in the applications of groupoids (cf. [ll]), and in this theory there are two key results. One is that if Q is a groupoid there is an equivalence between the category Or, (Q) of operations of Q on sets (or Q-sets aa they are called) and the category t70vlQ of covering groupoids of Q. (This result seems to have been stated first in this form in [9], ulthough the constructions involved had been used previously.) The other is t h a t if Q is a transitive groupoidi), there is a bijection between the equivalence olassee of transitive covering groupoids of Q and the conjugacy classes of subgroups of Q. (This result is an abstract formulation of known results on covering H;)acea and the fundamental groug-it is due to HIWINS [ll], page 110.)The object of this paper is t o prove topological versions of these results.For the first reeult there is no problem. We define the category XtfoulU of i~o~~ological covering morphisms of the topological groupoid Q ; we follow EHRES- MA"[7] in defining the category ZOr, (a) of &paces; and we prove the equiviilence of these categories. This allows us to give a number of useful examples in t.liese categories.The second problem presents difficulties which are related to the fact that a.I riuisitive Q-spaoe need not be a homogeneous space of Q. However we present a ctorresponding result for the locally trivial case. Jn general this paper is independent of [6]; however, we will not repeat any id' ihe results which appmr there.Rom? of the results of this paper appeared in [6] and [lo]. During part of.

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In this note the Stone-Cech compactification is used to produce short proofs of two theorems on the structure of free topological groups. The first is: The free topological group on any Tychonoff space X contains, as a closed subspace, a homeomorphic copy of the product space X". This is a generalization of a result of B. V. S. Thomas. The second theorem proved is C. Joiner's, Fundamental Lemma.

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