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We prove that each homogeneous continuum which topologically contains an n-dimensional unit cube and lies in (n + l)-dimensional Euclidean space is locally connected. Introduction. Although we still do not have any complete classification of homogeneous plane continua today, there are many significant results concerning homogeneous continua in the planar case. It is interesting to consider which of these results may be naturally extended to higher dimensional cases. In this paper we give an example of such an extension. In 1924 S. Mazurkiewicz [7] proved that a simple closed curve is the only homogeneous plane continuum which is nondegenerate and locally connected. Using this result, R. H. Bing showed in 1959 [1] that a simple closed curve is the only homogeneous plane continuum that contains an arc. All that he had to do to show this (and that he did) was to prove that each homogeneous plane continuum which contains an arc is locally connected. Here we extend the last statement to all finite dimensional cases. MAIN THEOREM. If a homogeneous continuum contains a set homeomorphic to the n-dimensional unit cube and lies in (n + 1)-dimensional Euclidean space, then it is locally connected. It is very natural to ask whether the assumptions of this theorem imply that the considered continuum is an n-manifold. Recently the author has answered this question in the affirmative. This result, which makes heavy use of the Main Theorem, will be presented in a future paper. The general method of the proof of the Main Theorem is similar to that of Bing's theorem, i.e. we suppose the existence of a continuum not satisfying this theorem and find a sequence of properties of it. Finally we get two properties such that one contradicts the other. However, only Property 1 here is similar to Property 4 in [1, p. 212], and the particular method of proof here differs much from Bing's. There is also another important difference from the Bing method we use many times the so-called s-push property of homogeneous metric continua (see Lemma 4 in [2, p. 37]) which is a corollary to the well-known Effros theorem, and which was not known to Bing in 1959. The author wishes to express profound thanks to Professor J. J. Charatonik for all his help in completing this work. Received by the editors October 15, 1986 and, in revised form, April 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20; Secondary 54C25.

Homogeneous continua in Euclidean (𝑛+1)-space which contain an 𝑛-cube are locally connected