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This paper revisits the notion of free configuration space and reviews some of its path-connectivity-related properties. The literature on motion planning reveals at least three different definitions for the free configuration space of a robot in the presence of obstacles. This paper shows that, assuming regularity of both object and obstacles, those three definitions are equivalent. We show that the three definitions regularize the free space and therefore prevent the existence of “thin bits,” or low-dimensional strata. The paper concludes by discussing a series of properties regarding the existence and smoothability of contact-free paths between pairs of configurations. [ABSTRACT FROM AUTHOR]