Elaboration of some ideas on identity path.

Question

In the book of Ralph H. Fox and Richard H.Crowell. The author said:

A path $a$ is called an identity path, if it has stopping time $||a|| = 0$. This terminology reflects the fact that the set of all identity paths in a topological space may be characterized as the set of all multiplicative identities with respect to the product. That is, the path $e$ is an identity iff $e . a = a$ and $b . e = b$ whenever $e . a$ and $b . e$ are defined.

Also the book said:

We call a path whose image is a single point a constant path. Every identity path is constant; but the converse is clearly false.

My questions are:

1-Is there a definition for the addition of 2 paths? and why identity paths in a topological space may be characterized as the set of multiplicative identities with respect to the product and not the set of additive identities with respect to the addition?

2-Is there a reason for putting $e$ on the left of $a$ and for putting it on the right of $b$ in the last line of the first paragraph?

3-According to the above definition of identity path, the loop is not an identity path even thought its stopping time is zero..... correct? and is the reason for not every constant path is an identity path, is that it may have the same image point for many inputs?

Could anyone help me in answering these questions please?

EDIT: A path $a$ in a topological space $X$ is then a continuous mapping $$ a: [0 , ||a||] \rightarrow X .$$ The number $||a||$ is the stopping time.

Answer 1

The product of paths is defined on p.14. If you multiply paths, then their stopping times (the lengths of the intervals on which they are defined) are added.

1. If you want, you can use instead of "multplication" the word "addition", but it is not recommendable since the word "addition" usually is understood as a commutative operation. Anyway, there is no other binary operation besides multiplication of paths.

2. Writing $e \cdot a$ and $b \cdot e$ only means that you can form both products - from the left and from the right. However, note that the occuring symbols $e$ are different identity paths although they have the same name. Perhaps it would be better to write $e_x$ for the identity path at a point $x$ ($e_x : [0,0] \to X, e_x(0) = x$). Doing so would lead to writing more precisely $e_{a(0)} \cdot a$ and $b \cdot e_{b(\lVert b \rVert)}$.

3. A loop is a path whose initial and terminal points coincide. Hence a loop is an identity path if and only if its stopping time is zero. All constant paths are loops, but again a constant path is an identity path if and onyl if its stopping time is zero.