The homology groups $H_n(X)$:

Definitions (29.3)-(29.6) and theorems (29.2)-(29.5) from the previous lecture show that given a topological space $X$, the sequence of groups $S_n(X)$ and group homomorphisms $\partial _n:S_n(X)\longrightarrow S_{n-1}(X)$ provide an example of a chain complex called the singular chain complex. If $f : X \longrightarrow Y$ is a continuous function, the sequence $f_{\sharp}:S_n(X)\longrightarrow S_n(Y)$ ( $n = 0, 1, 2,\dots$) defines a chain map from the chain complex $S(X)$ to $S(Y)$. The general results on chain complexes when applied to this special case gives us the homology functors from Top to AbGr.