I will summarize the results I developed and presented another thread in a more coherent and easier-to-follow fashion here.

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.