Proper noun
Brouwer fixed-point theorem
(topology) A theorem which (in one of its simplest forms) states that a continuous function from an n-dimensional unit cube (in
R
n
{\displaystyle \mathbb {R} ^{n}}
) to itself has a fixed point.
Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Brouwer fixed-point theorem for functions (or, more generally, the Kakutani fixed-point theorem for set-valued functions ). Source: Internet
Thurston accomplished his classification by observing that the mapping class group acted naturally on a compactification of Teichmüller space ; as this enlarged space was homeomorphic to a closed ball, the Brouwer fixed-point theorem became applicable. Source: Internet