Brouwer fixed-point theorem
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Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function f mapping a nonempty compact convex set to itself, there is
For flows in an unbounded area, or in an area with a "hole", the theorem is not applicable.
John Nash used the theorem in game theory to prove the existence of an equilibrium strategy profile.
Hairy ball theorem
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The hairy ball theorem of algebraic topology states that there is no non-vanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a contin
Brouwer fixed-point theorem
…its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used…
A continuous tangent vector field on a 2-sphere with only one pole, in this case a dipole field with index 2. See also an animated version of this graphic.
A hair whorl