##  [Borsuk–Ulam Theorem](/borsuk-ulam-theorem) 

  ##  [Borsuk–Ulam Theorem](https://mathlogic.quantumdictionary.io/borsuk-ulam-theorem-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The statement that any continuous map f from the n-sphere S^n to Euclidean n-space R^n maps some pair of antipodal points x and −x to the same image point: f(x)=f(−x).

 

 

 

 

 





 

 



 ##  [Borsuk–Ulam Theorem](https://puremath.quantumdictionary.io/borsuk-ulam-theorem-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A theorem in algebraic topology: any continuous map from the n-sphere S^n to Euclidean n-space R^n identifies at least one pair of antipodal points, i.e., there exists x in S^n with f(x)=f(−x).