I will summarize the results I developed and presented another thread in a more coherent and easier-to-follow fashion here.
You have two topological spaces X and Y and two continuous functions. Suppose for each , there corresponds a continuous function such that and for all .
A continuous functionis callled a homotopy from f to g. We say that f and g are homotopic iff there exists a homotopy from f to g, and we write ; we can also express the fact that H is a homotopy from f to g by writing .
The relation is homotopic to is an equivalence relation on, the class of all continuous functions from X to Y.
(i) For any, the function is a homotopy from f to itself.
(ii) For any, if , then is a homotopy from g to f.
(ii) For any, if and , define as follows.
the equivalence classes under this equivalence relation are called homotopy classes.
2. RELATIVE HOMOTOPY
Let X, Y be topological spaces,, and A a subset of X such that for all . If H is a homotopy from f to g such that for all , H is called a homotopy relative to A (or homotopy rel A) from f to g, and we can write .
Like ordinary homotopy, relative homotopy is an equivalence relation on. Indeed, if , H is just an ordinary homotopy.
Let X be a topological space and. A path in X from a to b is a continuous function with 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 pathis the path from a to c defined by
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...
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:
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..."
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.
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)