I. SummaryThis paper introduces the notion of an M-connected subset of a metric space and is concerned with approximation theoretic properties of subsets of finite dimensional real normed linear spaces. Let K denote a non-empty closed subset of a finite dimensional real normed linear space X. We are concerned with the three properties(Notation and terminology are presented systematically in Sect. 2. The letter M in "M-connected" modifies the word "connected", it is not a variable.) It is a particular case of a result of Vlasov [15, Theorem 2.5] that (2) implies (3). It is shown here (Theorem 3.6) that (1) implies (2). We introduce a class of spaces which we refer to as (BM)-spaces. The class of finite dimensional (BM)-spaces is closed under the formation of/~176 sums (Theorem 5.1) and contains every smooth space (Corollary 5.4) and every two-dimensional space the unit sphere of which is a polygon (Theorem 5.5). If X is either a two-dimensional space or a (BM)-space then (3) implies (1) (Theorems 4.1 and 4.2); thus in these spaces the three properties are equivalent. However, there exist spaces in which there are suns which are P-acyclic but not M-connected (Theorem 4.3). The final result of the paper (Theorem 5.7) identifies those finite dimensional (BM)-spaces, the unit spheres of which are polytopes.Koshcheev I-1(3] has shown that every sun in a finite dimensional space is connected. This is perhaps the only completely general result concerning suns in finite dimensional spaces; it can be compared with the fact that an M-connected closed subset of a finite dimensional space is path connected (Theorem 3.6(i)) and with the fact that a P-acyclic set in a finite dimensional space satisfies a very strong cohomological connectedness condition (in particular it is clc | -cohomologically locally connected) (Theorem 2.3).
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