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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

I will summarize the results I developed and presented another thread in a more coherent and easier-to-follow fashion here.

**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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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

**2. RELATIVE HOMOTOPY**

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

Like ordinary homotopy, relative homotopy is an equivalence relation on

. Indeed, if ,Offline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

**3. PATHS**

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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**LQ****Real Member**- Registered: 2006-12-04
- Posts: 1,285

I'm amazed, sounds important

I see clearly now, the universe have the black dots, Thus I am on my way of inventing this remedy...

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

In differential geometry, you also define the set of paths to a point to be the tangent space at that point. The paths are of the form:

Such that

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

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Thanks Ricky.

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.

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**Nehushtan****Member**- Registered: 2013-03-09
- Posts: 957

**5. HAM-SANDWICH THEOREM**

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.

*Last edited by Nehushtan (2016-04-21 00:30:28)*

**240** books currently added on Goodreads

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