Why is it that $\mathbb{Z}$ which has two generators, $1$ and $-1$, is isomorphic to $H_{0}$ of a line with two different vertices, "$\{x,y\} / <y - x >$"? Basically what are some elements not equal to zero so that I may compare them thanks -
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In a topological space $X$, $C_0(X)$ is the set of formal (finite with integer coefficients) linear combinations of points.
The image of $C_1(X)$ is the set of linear combinations of $[P]-[Q]$, where there exists a continuous path from $P$ to $Q$.
In other words, in a line, any point of the line corresponds to the same class in $H_0$, which is a generator of the group.
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