Path Components

Definition

Having a path between two points is an equivalence relation and the equivalence classes are called the path components of .
The set of path components is . If then is path connected.

Path component map

Proposition

Given a map , get a well define function:

and it satisfies:

2. If and are maps then

Proof

Firstly, well defined:

Find
Then so
(1) Say . Then for any we have a path , so .
The (2) and (3) are apparently directly from definitions.

Corollary

If is a homotopy equivalence then is a bijection.

Proof

Let be a homotopy inverse of .

Similarly for other way.

Example

, . Suppose there is a retraction with
Now , .
Also , but the first one is a bijection while the second one cannot be because .