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Armentrout [l] has shown that there is a Moore space on which every real-valued continuous function is constant. While this space is connected, it is not known to be complete. In answer to questions raised by F. B. Jones [2] it is shown in this paper that there is a connected, locally-connected, complete, separable Moore space on which every continuous real-valued function is constant. This space is very similar to one whose existence was announced by P. Roy [5]. A topological space is called a M\oore space if it satisfies Axiom 0 and the first three parts of Axiom 1 of [4]. A Moore space is said to be complete if it satisfies all of Axiom 1 in [4]. In a Moore space, domains are open sets and the set of regions whose existence is assured by Axiom 1 is a base for the topology. The construction uses a space which is essentially Armentrout's modification of a Moore space constructed by F. B. Jones [3]. It is redescribed here to make the construction more amenable to geonmetric intuition. The author is indebted to H. Cook for suggesting this geometric realization of the space 2. Let C be a planar disc topologized as follows. If P is on the rim of C, regions containing P shall be interiors of circles lying in C and having only P in common with the rim of C together with P. If P is in C but not on the rim of C, regions containing P shall be interiors of circles lying in C having P in their interior and containing no point of the rim of C. With this topology, C becomes a separable, connected, locally connected, complete Moore space S. The rim of C is a discrete set in S and is the union of two disjoint uncountable sets A and B with the property that any domain in S which contains an uncountable subset of one of them has an uncountable subset of the other in its closure [3]. Let M be a countable dense subset of S containing no point of the rim of C. Let T be the point set in E3 obtained by rotating the graph of y=x2-1, (-I

A locally connected, complete Moore space on which every real-valued continuous function is constant