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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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## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth