It is always possible to slice a three-layered ham sandwich with a single cut of a knife in such a way that each layer of the sandwich is divided into two exactly equal halves by the cut.

The ham-sandwich theorem can be proved using the Borsuk–Ulam theorem.

]]>Later, when I come to *n*-dimensional homotopy groups, I will be defining *n*-dimensional paths as continuous functions from the *n*-dimensional hypercube to *X* but for the time being I avoid jumping the gun.

Such that

Where M is your manifold. We then use the tangent space to define what a "derivative" means.

]]>Let *X* be a topological space and

If *p* is a path in *X* from *a* to *b* and *q* is a path in *X* from *b* to *c*, the product path

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Let *X*, *Y* be topological spaces,

Like ordinary homotopy, relative homotopy is an equivalence relation on

. Indeed, if ,**1. HOMOTOPY**

You have two topological spaces *X* and *Y* and two continuous functions

A continuous function

is callled a homotopy fromThe relation is homotopic to is an equivalence relation on

, the class of all continuous functions from(i) For any

, the function is a homotopy from(ii) For any

, if , then is a homotopy from(ii) For any

, if and , define as follows.Then

.the equivalence classes under this equivalence relation are called homotopy classes.

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