Topology/Homotopy

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Contents 1 Paths and Loops 1.1 Paths 1.2 Loops 2 Definition

[edit] Paths and Loops

We have worked with the concept of paths before in the concepts of path connectedness and local path connectedness. Here, we will review them again, and then define some new terms.

[edit] Paths

Definition: A path from x to y is a continuous function f from [0,1] to X such that f(0)=x and such that f(1)=y.

[edit] Loops

Definition: Let X be a topological space and . One says that α is a loop with base a if αis path from a to a.

[edit] Definition

Let X and Y be topological spaces, and let f(x) and g(x) be continuous functions from X to Y. A homotopy between f and g is a continuous function h(x,r) from the set X×[0,1] to Y, such that h(x,0)=f(x), and such that h(x,1)=g(x).

Intuitively, we can think of a homotopy between two functions as a kind of continuous mapping between the two functions.

One can easily verify that homotopy is an equivalence relation both on paths and loops.

Homotopy of loops:
We can define two loops to be homotopic when, when we consider a homotopy through loops i. e. a homotopy h(x,r) between the two loops such that h(x,0)=h(x,1) for all x.

Note: If then all loops with base a are homotopic. We just have to take

F(t,s) = (1 − s)α(t) + sβ(t).