Meaning of Brouwer fixed-point theorem

Brou'wer fixed'-point' the"orem

Pronunciation: (brou'ur fikst'point'), [key]
— Math. Math.
  1. the theorem that for any continuous transformation of a circle into itself, including its boundary, there is at least one point that is mapped to itself.
Random House Unabridged Dictionary, Copyright © 1997, by Random House, Inc., on Infoplease.