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 ![f:[0,1]->X](/images/equations/Arcwise-Connected/Inline4.svg)
such that 
and 
,
but must also have a continuous inverse function,
i.e., that it is a homeomorphism between ![[0,1]](/images/equations/Arcwise-Connected/Inline7.svg)
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 ![f:[0,1]->X](/images/equations/Arcwise-Connected/Inline10.svg)
defined by 
for all 
, and 
, is a path from 
to 
, but there exists no homeomorphism from ![[0,1]](/images/equations/Arcwise-Connected/Inline16.svg)
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).
