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family of sets in general position; convexly connected sets; Fan-Glicksberg-Kakutani fixed point theorem
In this paper the main result in [1], concerning $(n+1)$-families of sets in general position in ${\bold R}^n$, is generalized. Finally we prove the following theorem: If $\{A_1,A_2,\dots,A_{n+1}\}$ is a family of compact convexly connected sets in general position in ${\bold R}^n$, then for each proper subset $I$ of $\{1,2,\dots,n+1\}$ the set of hyperplanes separating $\cup\{A_i: i\in I\}$ and $\cup\{A_j: j\in \overline{I}\}$ is homeomorphic to $S_n^+$.
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