Editing Borsuk–Ulam theorem (section)

{{short description|Theorem in topology}}

{{technical|date=May 2025}}

[[File:Antipodal.png|thumb|right|alt=mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.|Antipodal]]
In [[mathematics]], the '''Borsuk–Ulam theorem''' states that every [[continuous function]] from an [[n-sphere|''n''-sphere]] into [[Euclidean space|Euclidean ''n''-space]] maps some pair of [[antipodal point]]s to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

Formally: if <math>f: S^n \to \R^n</math> is continuous then there exists an <math>x\in S^n</math> such that:  <math>f(-x)=f(x)</math>.

The case <math>n=1</math> can be illustrated by saying that there always exist a pair of opposite points on the [[Earth]]'s equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space, which is, however, not always the case.<ref name=":0">{{Cite journal |last1=Jha |first1=Aditya |last2=Campbell |first2=Douglas |last3=Montelle |first3=Clemency |last4=Wilson |first4=Phillip L. |date=2023-07-30 |title=On the Continuum Fallacy: Is Temperature a Continuous Function? |journal=Foundations of Physics |language=en |volume=53 |issue=4 |pages=69 |doi=10.1007/s10701-023-00713-x |issn=1572-9516|doi-access=free |hdl=1721.1/152272 |hdl-access=free }}</ref>

The case <math>n=2</math> is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space.  

The Borsuk–Ulam theorem has several equivalent statements in terms of [[odd function]]s. Recall that <math>S^n</math> is the [[n-sphere|''n''-sphere]] and <math>B^n</math> is the [[n-ball|''n''-ball]]:
* If <math>g : S^n \to \R^n</math> is a continuous odd function, then there exists an <math>x\in S^n</math> such that:  <math>g(x)=0</math>.
* If <math>g : B^n \to \R^n</math> is a continuous function which is odd on <math>S^{n-1}</math> (the boundary of <math>B^n</math>), then there exists an <math>x\in B^n</math> such that:  <math>g(x)=0</math>.