Tychonoff cube
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. Definition. Let formula_1 denote the unit interval formula_2. Given a cardinal number formula_3, we define a Tychonoff cube of weight formula_4 as the space formula_5 with the product topology, i.e. the product formula_6 where formula_7 is the cardinality of formula_8 and, for all formula_9, formula_10. The Hilbert cube, formula_11, is a special case of a Tychonoff cube. Properties. The axiom of choice is assumed throughout.