Borsuk-Ulam theorem

Call a continuous map $f:S^{m}\to S^{n}$ antipode preserving if $f(-x)=-f(x)$ for all $x\in S^{m}$ .

Theorem: There exists no continuous map $f:S^{n}\to S^{n-1}$ 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 $\mathbb{R}^{n}$ is homeomorphic to $S^{n}$ .

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