Reduced homology of wedge sum relative to basepoints with neighborhoods that deformation retract to them is direct sum of reduced homologies

For two spaces
Suppose $$(X,x_0)$$ and $$(Y,y_0)$$ are based topological spaces. Define the wedge sum:

$$\! (Z,z_0) = (X,x_0) \vee (Y,y_0)$$

Here $$Z = (X \sqcup Y)/\{x_0,y_0 \}$$ and $$z_0$$ is the identified point $$\! x_0 \sim y_0$$.

Suppose that, in $$(X,x_0)$$, there exists an open subset $$A$$ containing $$x_0$$ that has a strong deformation retraction to $$x_0$$. Suppose that in $$(Y,y_0)$$, there exists an open subset $$B$$ containing $$y_0$$ that has a strong deformation retraction to $$y_0$$. Then, for all nonnegative integers $$k$$, we have the following isomorphism for reduced homology groups:

$$\! \tilde{H}_k(Z) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$$

Note that for $$k > 0$$, this gives an isomorphism of homology groups:

$$\! H_k(Z) \cong H_k(X) \oplus H_k(Y)$$

For $$k = 0$$, we get:

$$\! H_k(Z) \oplus \mathbb{Z} cong H_k(X) \oplus H_k(Y)$$

where $$\mathbb{Z}$$ is replaced by the module of coefficients for homology.

Related facts

 * Fundamental group of wedge sum relative to basepoints with neighborhoods that deformation retract to them is free product of fundamental groups

Facts used

 * 1) uses::Mayer-Vietoris homology sequence

Proof
Given: Based topological spaces $$(X,x_0)$$ and $$(Y,y_0)$$. There is a contractible open subset $$A$$ of $$X$$ containing $$x_0$$ and a contractible open subset $$B$$ of $$Y$$ containing $$y_0$$. $$(Z,z_0) \ (X,x_0) \vee (Y,y_0)$$.

To Prove: $$\tilde{H}_k(Z) \cong \tilde{H}_k(X) \oplus \tilde{H}_k(Y)$$.

Proof: We use the Mayer-Vietoris homology sequence, taking the following open subsets $$U$$ and $$V$$ of $$Z$$:

$$\! U = A \cup Y$$

and:

$$\! V = X \cup B$$