Arcwise-Connected
المؤلف:
Armstrong, M. A.
المصدر:
Basic Topology, rev. ed. New York: Springer-Verlag
الجزء والصفحة:
...
14-7-2021
4417
Arcwise-Connected
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](https://mathworld.wolfram.com/images/equations/Arcwise-Connected/Inline4.gif)
such that 
and 
, but must also have a continuous inverse function, i.e., that it is a homeomorphism between ![[0,1]](https://mathworld.wolfram.com/images/equations/Arcwise-Connected/Inline7.gif)
and the image of 
.
The difference between the two notions can be clarified by a simple example. The set ![X=<span style=]()
{a,b}" src="https://mathworld.wolfram.com/images/equations/Arcwise-Connected/Inline9.gif" style="height:15px; width:60px" /> with the trivial topology is pathwise-connected, but not arcwise-connected since the function ![f:[0,1]->X](https://mathworld.wolfram.com/images/equations/Arcwise-Connected/Inline10.gif)
defined by 
for all 
, and 
, is a path from 
to 
, but there exists no homeomorphism from ![[0,1]](https://mathworld.wolfram.com/images/equations/Arcwise-Connected/Inline16.gif)
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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