Many authors (e.g., Mendelson 1963; Pervin 1964) use the term arcwise-connected as a synonym for pathwise-connected. Other authors (e.g., Armstrong 1983; Cullen 1968; and Kowalsky 1964) use the term to refer to a stronger type of connectedness, namely that an arc connecting two points a and b of a topological space X is not simply (like a path) a continuous function f:[0, 1]->X such that f(0) = a and f(1) = b, but must also have a continuous inverse function, i.e., that it is a homeomorphism between [0, 1] and the image of f.

connected set | continuous | homeomorphism | locally pathwise-connected | pathwise-connected | trivial topology