Homology of wedge sum is the direct sum of homologies

I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a neighborhood of $A$ that deformation retracts onto $A$).

What I tried:

Since $(X_\alpha, x_\alpha)$'s are good pairs, $(\bigsqcup X_\alpha, \{x_\alpha:\alpha\in I\})$ is a good pair, so a theorem (a long exact sequence argument) gives us an isomorphism $$q_*:H_n(\bigsqcup X_\alpha, \{x_\alpha:\alpha\in I\})\to H_n(\bigsqcup X_\alpha/\{x_\alpha:\alpha\in I\},\{x_\alpha:\alpha\in I\}/\{x_\alpha:\alpha\in I\})=H_n(\bigvee X_\alpha, \text{some point})$$ induced by the quotient map $$q:\bigsqcup X_\alpha, \{x_\alpha:\alpha\in I\})\to \bigsqcup X_\alpha/\{x_\alpha:\alpha\in I\},\{x_\alpha:\alpha\in I\}/\{x_\alpha:\alpha\in I\})=(\bigvee X_\alpha, \text{some point})$$

Questions:

Now, do these mean that we have an isomorphism $\phi:H_n(\bigsqcup X_\alpha)=\bigoplus H_n(X_\alpha) \to H_n(\bigvee X_\alpha)$? In general if we have an isomorphism $\theta:H_n(X,Y)\to H_n(A,B)$, then do we also have an isomorphism $\theta':H_n(X)\to H_n(A)$?

Your proof is almost complete, let me suggest you some additional hint to complete it:

• there's an isomorphism $$\tilde H_n(X) \cong H_n(X,x_0)$$ between the reduced homology of a space $X$ and the homology of the pair $(X,x_0)$ where $x_0 \in X$;

• in the category of pairs of topological spaces there's an isomorphism $$H_n\left(\bigsqcup_\alpha (A_\alpha,B_\alpha)\right) \cong \bigoplus_\alpha H_n(A_\alpha, B_\alpha)$$ for a family of pairs $(A_\alpha,B_\alpha)_\alpha$.

Combining these results with what you used should bring you to the solution of the problem.

Hope this helps.