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.

The difference between the two notions can be clarified by a simple example. The set X={a,b} with the trivial topology is pathwise-connected, but not arcwise-connected since the function f:[0,1]->X defined by f(t)=a for all t!=1, and f(1)=b, is a path from a to b, but there exists no homeomorphism from [0,1] to X, 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).

See also

Connected, Connected Set, Continuous, Homeomorphism, Locally Pathwise-Connected, Pathwise-Connected, Trivial Topology

This entry contributed by Margherita Barile

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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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Barile, Margherita. "Arcwise-Connected." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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