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 and of a topological space is not simply (like a path) a
continuous function
such that
and ,
but must also have a continuous inverse function,
i.e., that it is a homeomorphism between and the image of .

The difference between the two notions can be clarified by a simple example. The set
with the trivial topology is pathwise-connected,
but not arcwise-connected since the function defined by for all , and , is a path from to , but there exists no homeomorphism from to , since even injectivity is
impossible.

Arcwise- and pathwise-connected are equivalent in Euclidean spaces and in all topological spaces having a sufficiently rich structure. In particular theorem states that every locally compact, connected, locally connected metrizable topological space is arcwise-connected (Cullen 1968, p. 327).