A Helly property of arcs (Q1109340)

If \(n+1\) arcs are given in the n-cube, each joining (0,...,0) to (1,...,1), then there is an arc joining the same points, and which is in between each n of the given arcs. The result is false for less than \(n+1\) arcs. The involved concept of betweenness is a natural one and relates with a convexity on the space of all arcs. The above result (which will be derived for n-dimensional compact arcwise connected Heyting algebras) can be compared with the classical Helly Theorem for Euclidean space.