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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 correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

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new correction: Many corrections by Smarandache

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new question: how to contest an entry? by zorba

new question: simple question by parag

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