How to see what the standard simplices of a topological space are? I am studying algebraic topology from Allen Hatcher´s book.
I understand the theory, but have trouble computing simplicial homology for spaces.
The problem for me is to see how the standard simplices look, based on a picture of the given space.
Definitions: A simplex $\Delta_n$ is defined (in Hatcher´s book) as the join of $n$ points is a convex polyhedron of dimension n − 1.
$\Delta_n(X)$ is then defined as the free abelian group with basis the open $n$-simplices $e^n_\alpha$ of $X$ . Elements of $\Delta_n(X)$ are called n-chains and can be written as finite formal sums $\sum_\alpha n_\alpha e^n_\alpha$ with coefficients $n_\alpha \in \mathbb{Z}$.
For example, calculating simplicial homology of $S^1$, we need to describe $\Delta_0(S^1)$ and $\Delta_1(S^1)$. As far as I understand, $\Delta_0(S^1)$ should be a discrete set of 0-cells (points) and $\Delta_1(S^1)$ should be a set of lines. But I don´t see how they are equal to $\mathbb{Z}$?
This is the example in Hatcher´s book I am referring to:

 A: Hatcher is a bit misleading in the chapter 2.1. What he considers are not "raw" simplices. In fact the circle $S^1$ contains no "raw" simplices of dimension above $0$ (e.g. no straight line is a subset of a circle). Instead he considers a $\Delta$-complex structure on $X$ (which is defined at the begining of the chapter), that is a collection of maps $\Delta^n\to X$ which satisfies certain properties. This can be seen when he says:

Example 2.2. $X=S^1$ with one vertex $v$ and one edge $e$.

He means one $0$-dimensional simplex $\Delta^0\to S^1$ and one $1$-dimensional simplex $\Delta^1\to S^1$ (the latter is a glueing of its endpoints). Both together form a $\Delta$-complex on $S^1$. Of course this depends on the choice of those maps, so these are fixed implicitly. Also Hatcher will use the word "simplex" or "vertex" or "edge" for such maps, and sometimes even for images of those maps as well. Yeah, it can be confusing.
He then builds $\Delta_n(X)$ chain complexes. Those complexes can be built on top of the $\Delta$-structure. Hatcher describes how it is done in the chapter as well.

$\Delta_0(S^1)$ should be a discrete set of 0-cells (points) and $\Delta_1(S^1)$ should be a set of lines

No. $\Delta_0(S^1)$ is a free group generated by all $0$-simplexes. But not all $0$-simplexes taken from thin air. All $0$-simplexes taken from a fixed $\Delta$-complex structure. In our case this is the single $\Delta^0\to S^1$ map. Hence it is $\mathbb{Z}$. Analogously $\Delta_1(S^1)$ is a free group generated by all $1$-simplexes. In our case this is the single $\Delta^1\to S^1$ map. Thus again $\mathbb{Z}$.
