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For a disjoint union $\sqcup_{i \in I} X_i$ of spaces $X_i$, the $n$th homology group is the direct sum of the homology groups of the pieces: $H_n(\sqcup_{i \in I} X_i) = \bigoplus_{i \in I} H_n(X_i)$ for all $n$. This axiom allows us to compute homology for disjoint unions piecewise.

For a subset $A \subseteq X$ whose closure is contained in the interior of another subset $B \subseteq X$, the inclusion $(X \setminus A, B \setminus A) \hookrightarrow (X,B)$ induces an isomorphism in homology: $H_n(X \setminus A, B \setminus A) \cong H_n(X,B)$ for all $n$. This axiom shows that the homology depends only on the local structure, not the set $A$ itself.

For a one-point space $\{pt\}$, the $n$th homology group is trivial for all $n>0$: $H_n(\{pt\}) = 0$ for $n>0$ and $H_0(\{pt\}) = \mathbb{Z}$. This axiom establishes the initial case for building up the theory of homology groups.

For any pair $(X,A)$, where $A$ is a subspace of $X$, there is a long exact sequence

If $f: (X,A) \rightarrow (Y,B)$ is a continuous map, then it induces a homomorphism $f_*: H_n(X,A) \rightarrow H_n(Y,B)$ for each $n$, which aligns with composition of functions ($(g \circ f)_* = g_* \circ f_*$). This axiom ensures that homology behaves predictably under continuous maps.

For a one-point space $\{pt\}$, the only non-trivial homology group is $H_0(\{pt\}) = \mathbb{Z}$, which indicates that we can 'begin' homology theory with a simple familiar object, the point.