Singular homology group

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

Let $X$ be a topological space and $A$ an abelian group. The $n$ -th singular homology group of $X$ with coefficients in $A$ is

H_n(X;A):=H_n(C_\bullet(X;A)) =\ker(\partial_n)/\operatorname{im}(\partial_{n+1}),

where $C_\bullet(X;A)$ is the singular chain complex. Its elements are cycles modulo boundaries.

Functoriality

A continuous map $f:X\to Y$ induces $f_*:H_n(X;A)\to H_n(Y;A)$ , and homotopic maps induce the same map. Thus singular homology is a covariant homotopy invariant.

Reduced and relative forms

The reduced group $\widetilde H_n(X;A)$ modifies degree zero using the augmentation to $A$ . For a pair $B\subseteq X$ , the relative group is defined from the quotient chain complex and is recorded separately as relative singular homology.

Reference

See Allen Hatcher, Algebraic Topology, Chapter 2, author-hosted book.