# Borsuk’s theorem

Borsuk-Ulam theorem. In mathematics , particularly in algebraic topology , there is a famous theorem conjectured by Stanisław Ulam and proved, first time, by Karol Borsuk in 1933 , dealing with antipodal points of a topological space and continuous functions .

It is noted that an n-sphere is defined as:

n = {(x 1 , x 2 , …, x n + 1 )}

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• 1 Theorem 1
• 2 Theorem 2
• 3 Theorem 3
• 4 Teroema 4
• 5 Source

## Theorem 1

Let b: S n → S n be a continuous application that transforms antipodal points into antipodal points; that is, for every element x of S n , bA (x) = Ab (x). So the Lefschetz number is an even integer.

## Theorem 2

Let b: S n → S n (n> 0) such that it assigns antipodal points to antipodal points. So tr [b * , H 0 (S n )] is an even number .

## Theorem 3

There is no application b: S n → S n (n> 0) that transforms antipodal points into antipodal points.

## Teroemma 4

Any continuous application of S n to R n assigns a pair of antipodal points at the same point. This last proposition bears the compound name of Borsuk-Ulam.

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