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).
REFERENCES:
Armstrong, M. A. Basic Topology, rev. ed. New York: Springer-Verlag, p. 112, 1997.
Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, pp. 325-330, 1968.
Kowalsky, H. J. Topological Spaces. New York: Academic Press, p. 183, 1964.
Mendelson, B. Introduction to Topology. London, England: Blackie & Son, 1963.
Pervin, W. J. "Arcwise Connectivity." §4.5 in Foundations of General Topology. New York: Academic Press, pp. 67-68, 1964.
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