Defining the term "connected" Some authors use the word "connected" as applied to a set S to mean that for any two points A and B in S there exits a (piecewise) smooth path in S from A to B. This is, for instance, the approach taken by Apostol in Calculus Vol II (p 332) and the approach taken by C.H. Edwards in Advanced Calculus of Several Variables (p 84) On the other hand, topology texts generally define the term "connected" to just mean that S cannot be expressed as two disjoint nonempty subsets and use the term "path connected" to indicate that any two points can be joined by a path (i.e., a continuous map from a closed interval into S) that lies entirely in S. (E.G, Munkres) So, then, what Apostol and Edwards mean is that S is "differentiably path connected" or "smoothly path connected". Is there established (updated?) terminology that identifies a space as "differentiably path connected" in the sense that these authors intend? In other words, a space in which any two points can be joined by a smooth path that lies in the space is called ______ ? Thanks.