In searching the world for examples of interesting topological objects, it may not be obvious that an outstanding place to look is within the walls of an automated warehouse or factory. In this exposition we describe a class of topological spaces that arise naturally in this very context. The examples we construct actually arose simultaneously in two seemingly disparate fields: the first author, in his thesis [1], discovered these spaces after working on problems about random walks on graphs with H. Landau, Z. Landau, J. Pommersheim, and E. Zaslow [2]. The second author discovered these same spaces while collaborating with D. Koditschek in the Artificial Intelligence Lab at the University of Michigan [7], [8]. In Sections 1 and 2 we give some motivations arising from robotics, as well as a little background on configuration spaces. For the remainder of the paper we focus on a fascinating class of topological spaces related to motion-planning on graphs.
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