Borsuk-Ulam theorem

Call a continuous mapMathworldPlanetmath f:SmSn antipode preserving if f(-x)=-f(x) for all xSm.

Theorem: There exists no continuous map f:SnSn-1 which is antipode preserving for n>0.

Some interesting consequences of this theorem have real-world applications. For example, this theorem implies that at any time there exists antipodal points on the surface of the earth which have exactly the same barometric pressure and temperature.

It is also interesting to note a corollary to this theorem which states that no subset of n is homeomorphic to Sn.

Title Borsuk-Ulam theorem
Canonical name BorsukUlamTheorem
Date of creation 2013-03-22 12:00:18
Last modified on 2013-03-22 12:00:18
Owner RevBobo (4)
Last modified by RevBobo (4)
Numerical id 7
Author RevBobo (4)
Entry type Theorem
Classification msc 54C99
Related topic HamSandwichTheorem