Cantor's Discontinuum

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

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.