Editing Borsuk–Ulam theorem (section)

====Algebraic topological proof====
Assume that <math>h: S^n \to S^{n-1}</math> is an odd continuous function with <math>n > 2</math> (the case <math>n = 1</math> is treated above, the case <math>n = 2</math> can be handled using basic [[Covering space|covering theory]]). By passing to orbits under the antipodal action, we then get an induced continuous function <math>h': \mathbb{RP}^n \to \mathbb{RP}^{n-1}</math> between [[Real projective space|real projective spaces]], which induces an isomorphism on [[Fundamental group|fundamental groups]]. By the [[Hurewicz theorem]], the induced [[ring homomorphism]] on [[cohomology]] with <math>\mathbb F_2</math> coefficients [where <math>\mathbb F_2</math> denotes the [[GF(2)|field with two elements]]], 

:<math> \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},</math>

sends <math>b</math> to <math>a</math>. But then we get that <math>b^n = 0</math> is sent to <math>a^n \neq 0</math>, a contradiction.<ref name=rotman>Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag {{ISBN|0-387-96678-1}} ''(See Chapter 12 for a full exposition.)''</ref>

One can also show the stronger statement that any odd map <math>S^{n-1} \to S^{n-1}</math> has odd [[degree of a continuous mapping|degree]] and then deduce the theorem from this result.