The space of $S^1/S^1$, the space of a single point, and their first homotopy group I read from the book Soft matter physics by Kleman that the space $R$ of a point is $0$ and its first homotopy group $\pi_1(0)=0$. This causes some confusion to my understanding. 


*

*Why the space of a single point is $0$ rather than $1$? I consider there is still one element in the space.

*Is the space of $S^1/S^1$ a single point? If so, is $S^1/S^1 = 1$ or $0$?

*Is $\pi_1(0) = 0$?


I am learning some basic topology at the same time (from the physics side of view), but I still only know a few of them. Can any give me some hints? Any hits are warmly welcome.
 A: Well, most of these questions are pretty much about notation, which I love/hate getting in to discussions about.  Having said that, I've rarely seen the space of a point denoted by a numeral, but when I have it's been $1$.  Somewhat less rarely, I've seen the group with one element denoted by a numeral, and almost always as $1$, unless it's clear the group is abelian.  So, if we denote both as $1$, then I could see writing $\pi_1(1)=1$, but that looks completely alien to me. 
To answer the more serious questions in your post, yes $S^1/S^1$ is a one point space, and yes the fundamental group of a point is the trivial group of one element.  These should be obvious from the definitions, so I think your understanding of quotient spaces and fundamental groups is pretty fuzzy, not helped by reading from a book using ( to me ) unusual notation.  
You can think of the quotient $X/Y$ as formed by gluing all the points of $Y$ to one point.  So, in this case, it should be obvious that $S^1/S^1$ is a single point.  
$\pi_1$ of a space is the collection of continuous maps from $S^1$ to that space, with some maps identified with each other ( and more structure that makes it a group,  etc. ).  In the case of a one point space, there is only one map from $S^1$ to that space, so regardless of the rest of the definition of $\pi_1$, it only has one element, so yes, $\pi_1$ of a one point space is the one element group. 
