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# 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 $S^{{n}}$.

Related:

HamSandwichTheorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

54C99*no label found*

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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

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new image: plot W(t) = P(waiting time <= t) by robert_dodier