Topologist's Sine Curve


An example of a subspace of the Euclidean plane that is connected but not pathwise-connected with respect to the relative topology. It is formed by the ray y=0, x<=0 and the graph of the function f(x)=sin(1/x) for x>0. This set contains no path connecting the origin with any point on the graph.

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This entry contributed by Margherita Barile

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Joshi, K. D. Introduction to General Topology. New Delhi, India: Wiley, pp. 151-153, 1983.Willard, S. General Topology. Reading, MA: Addison-Wesley, p. 198, 1970.

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Topologist's Sine Curve

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Barile, Margherita. "Topologist's Sine Curve." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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