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In this paper, certain separation theorems, which hold in spaces satisfying R. L. Moore's Axioms 0 and 1-5 of [2], are established for a class of spaces in which there do not necessarily exist either arcs or simple closed curves. The spaces considered satisfy axioms suggested by those of Moore. Consider a nondegenerate complete metric space S which satisfies the following axioms from [2]: AxioM 2. S is locally connected. AXIoM 3. S is connected and has no cut point. AXIoM 4. If J is a simple closed curve in S, then S J has exactly two components and J is the boundary of each of them. AXIoM 5. If x and y are distinct points and U is an open set containing x, then there is a simple closed curve J in U such that J separates x from y. It will be shown that even with a weakening of these conditions on S, the following separation theorems hold: (1) If neither of the compact point sets H and K separates the two points x and y, and either H and K are disjoint or HOK is connected, then HkJK does not separate x and y. (2) If the common part of the two compact continua H and K is not connected, then HUK separates S. Principally, the conditions above are weakened by the omission of the stipulation that S be complete. It is assumed instead that S is locally compactly connected (a point set K in S is compactly connected if and only if each two points of K belong to a compact continuum in K). As an example in ?4 shows, in a locally compactly connected metric space, there does not necessarily exist an arc. However, there exist chains of compact continua, and sets which are the unions of links of such chains serve as substitutes for arcs. The sets which serve as substitutes for simple closed curves are the unions of the links of closed chains of compact continua. Moore's Axioms 4 and 5 are replaced by analogous axioms for closed chains. This problem was suggested to the writer by R. L. Moore, and the development presented here is along lines similar to that in [2] which leads to the separation theorems mentioned above. 1. Consequences of Axioms 1 and 2. AXIOM 1. S is a nondegenerate connected metric space with no cut point. The diameter of a bounded point set M and the distance between two point sets H and K are denoted by diam M and dist (H, K), respectively.

Separation theorems for some plane-like spaces