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
. 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 path
is the path from a to c defined by]]>
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.]]>1. HOMOTOPY
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 function
is 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.Then
.the equivalence classes under this equivalence relation are called homotopy classes.
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