[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/borsuk-ulam-theorem-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/borsuk-ulam-theorem-wikipedia\/","headline":"Borsuk\u2013Ulam theorem – Wikipedia","name":"Borsuk\u2013Ulam theorem – Wikipedia","description":"before-content-x4 Theorem in topology after-content-x4 In mathematics, the Borsuk\u2013Ulam theorem states that every continuous function from an n-sphere into Euclidean","datePublished":"2019-07-21","dateModified":"2019-07-21","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2d6c446e22a8bf3bff82045f5ad674a9397ae57d","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2d6c446e22a8bf3bff82045f5ad674a9397ae57d","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/borsuk-ulam-theorem-wikipedia\/","wordCount":12351,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Theorem in topology       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, the Borsuk\u2013Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points 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 f:Sn\u2192Rn{displaystyle f:S^{n}to mathbb {R} ^{n}} is continuous then there exists an        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4x\u2208Sn{displaystyle xin S^{n}} such that:  f(\u2212x)=f(x){displaystyle f(-x)=f(x)}.The case n=1{displaystyle n=1} 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.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The case n=2{displaystyle n=2} 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\u2013Ulam theorem has several equivalent statements in terms of odd functions. Recall that Sn{displaystyle S^{n}} is the n-sphere and Bn{displaystyle B^{n}} is the n-ball:Table of ContentsHistory[edit]Equivalent statements[edit]With odd functions[edit]With retractions[edit]1-dimensional case[edit]General case[edit]Algebraic topological proof[edit]Combinatorial proof[edit]Corollaries[edit]Equivalent results[edit]Generalizations[edit]See also[edit]References[edit]External links[edit]History[edit]According to Matou\u0161ek (2003, p.\u00a025), the first historical mention of the statement of the Borsuk\u2013Ulam theorem appears in Lyusternik & Shnirel’man (1930).  The first proof was given by Karol Borsuk\u00a0(1933), where the formulation of the problem was attributed to Stanislaw Ulam.  Since then, many alternative proofs have been found by various authors, as collected by Steinlein (1985).Equivalent statements[edit]The following statements are equivalent to the Borsuk\u2013Ulam theorem.[1]With odd functions[edit]A function g{displaystyle g} is called odd (aka antipodal or antipode-preserving) if for every x{displaystyle x}: g(\u2212x)=\u2212g(x){displaystyle g(-x)=-g(x)}.The Borsuk\u2013Ulam theorem is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero. PROOF:With retractions[edit]Define a retraction as a function h:Sn\u2192Sn\u22121.{displaystyle h:S^{n}to S^{n-1}.} The Borsuk\u2013Ulam theorem is equivalent to the following claim: there is no continuous odd retraction.Proof: If the theorem is correct, then every continuous odd function from Sn{displaystyle S^{n}} must include 0 in its range. However,  0\u2209Sn\u22121{displaystyle 0notin S^{n-1}} so there cannot be a continuous odd function whose range is Sn\u22121{displaystyle S^{n-1}}.Conversely, if it is incorrect, then there is a continuous odd function g:Sn\u2192Rn{displaystyle g:S^{n}to {mathbb {R}}^{n}} with no zeroes. Then we can construct another odd function h:Sn\u2192Sn\u22121{displaystyle h:S^{n}to S^{n-1}} by:h(x)=g(x)|g(x)|{displaystyle h(x)={frac {g(x)}{|g(x)|}}}since g{displaystyle g} has no zeroes, h{displaystyle h} is well-defined and continuous. Thus we have a continuous odd retraction.1-dimensional case[edit]The 1-dimensional case can easily be proved using the intermediate value theorem (IVT).Let g{displaystyle g} be an odd real-valued continuous function on a circle. Pick an arbitrary x{displaystyle x}. If g(x)=0{displaystyle g(x)=0} then we are done. Otherwise, without loss of generality, 0.}”\/> But g(\u2212x)Sn\u22121{displaystyle h:S^{n}to S^{n-1}} is an odd continuous function with 2″\/> (the case n=1{displaystyle n=1} is treated above, the case n=2{displaystyle n=2} can be handled using basic covering theory). By passing to orbits under the antipodal action, we then get an induced continuous function h\u2032:RPn\u2192RPn\u22121{displaystyle h’:mathbb {RP} ^{n}to mathbb {RP} ^{n-1}} between real projective spaces, which induces an isomorphism on fundamental groups. By the Hurewicz theorem, the induced ring homomorphism on cohomology with F2{displaystyle mathbb {F} _{2}} coefficients [where F2{displaystyle mathbb {F} _{2}} denotes the field with two elements],F2[a]\/an+1=H\u2217(RPn;F2)\u2190H\u2217(RPn\u22121;F2)=F2[b]\/bn,{displaystyle mathbb {F} _{2}[a]\/a^{n+1}=H^{*}left(mathbb {RP} ^{n};mathbb {F} _{2}right)leftarrow H^{*}left(mathbb {RP} ^{n-1};mathbb {F} _{2}right)=mathbb {F} _{2}[b]\/b^{n},}sends b{displaystyle b} to a{displaystyle a}. But then we get that bn=0{displaystyle b^{n}=0} is sent to an\u22600{displaystyle a^{n}neq 0}, a contradiction.[2]One can also show the stronger statement that any odd map Sn\u22121\u2192Sn\u22121{displaystyle S^{n-1}to S^{n-1}} has odd degree and then deduce the theorem from this result.Combinatorial proof[edit]The Borsuk\u2013Ulam theorem can be proved from Tucker’s lemma.[1][3][4]Let g:Sn\u2192Rn{displaystyle g:S^{n}to mathbb {R} ^{n}} be a continuous odd function. Because g is continuous on a compact domain, it is uniformly continuous. Therefore, for every "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/borsuk-ulam-theorem-wikipedia\/#breadcrumbitem","name":"Borsuk\u2013Ulam theorem – Wikipedia"}}]}]