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Cantor's Discontinuum

A Cartesian product of any finite or infinite set I of copies of Z_2, equipped with the product topology derived from the discrete topology of Z_2. It is denoted Z_2^I. The name is due to the fact that for I=N, this set is closely related to the Cantor set (which is formed by all numbers of the interval [0,1] which admit an expansion in base 3 formed by 0s and 2s only), and this gives rise to a one-to-one correspondence between Z_2^N and the Cantor set, which is actually a homeomorphism. In the symbol denoting the Cantor discontinuum, Z_2 can be replaced by 2 and N by aleph_0.

See also

Cantor Set

This entry contributed by Margherita Barile

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References

Cullen, H. F. "The Cantor Ternary Space." §18 in Introduction to General Topology. Boston, MA: Heath, pp. 77-81, 1968.Joshi, K. D. Introduction to General Topology. New Delhi, India: Wiley, p. 199, 1983.Willard, S. "The Cantor Set." §30 in General Topology. Reading, MA: Addison-Wesley, pp. 216-219, 1970.

Referenced on Wolfram|Alpha

Cantor's Discontinuum

Cite this as:

Barile, Margherita. "Cantor's Discontinuum." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CantorsDiscontinuum.html

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