Editing Tychonoff's theorem (section)

== Applications ==

Tychonoff's theorem has been used to prove many other mathematical theorems. These include theorems about compactness of certain spaces such as the [[Banach–Alaoglu theorem]] on the weak-* compactness of the unit ball of the [[dual space]] of a [[normed vector space]], and the [[Arzelà–Ascoli theorem]] characterizing the sequences of functions in which every subsequence has a [[uniform convergence|uniformly convergent]] subsequence. They also include statements less obviously related to compactness, such as the [[De Bruijn–Erdős theorem (graph theory)|De Bruijn–Erdős theorem]] stating that every [[critical graph|minimal ''k''-chromatic graph]] is finite, and the [[Curtis–Hedlund–Lyndon theorem]] providing a topological characterization of [[cellular automaton|cellular automata]].

As a rule of thumb, any sort of construction that takes as input a fairly general object (often of an algebraic, or topological-algebraic nature) and outputs a compact space is likely to use Tychonoff: e.g., the [[Gelfand representation|Gelfand space]] of maximal ideals of a commutative [[C*-algebra]], the [[Stone space]] of maximal ideals of a [[Boolean algebra (structure)|Boolean algebra]], and the [[Berkovich spectrum]] of a commutative [[Banach ring]].