Paths

Definition (path)

Let be a space with . A path from to is a map st and .
If then is a loop based at .

Notation

Definition (concatenation)

Let and . Then we define a new path

Note this is OPPOSITE to function composition.

Definition (inverse)

with

Definition (constant path)

Definition (homotopic paths)

Paths and are homotopic as paths from to if they are homotopic as maps relative to

Path Components