Tychonoff cube

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In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tychonoff cube is named after Andrey Tychonoff, who first considered the arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact. Tychonoff later generalized this to the product of collections of arbitrary compact spaces. This result is now known as Tychonoff's theorem and is considered one of the most important results in general topology.[1]

Definition[edit]

Let  I denote the unit interval [0,1]. Given a cardinal number \kappa \geq \omega, we define a Tychonoff cube of weight \kappa as the space  I^\kappa with the product topology, i.e. the product  \prod_{s\in S} I_s where \kappa is the cardinality of  S and, for all  s\in S ,  I_s = I .

The Hilbert cube,  I^\omega , is a special case of a Tychonoff cube.

Properties[edit]

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References[edit]

Notes[edit]

  1. ^ Willard, Stephen (2004), General Topology, Mineola, NY: Dover Publications, ISBN 0-486-43479-6