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

Statement
Suppose $$(X,x_0)$$ and $$(Y,y_0)$$ are based topological spaces. Suppose, further, that $$X$$ and $$Y$$ are both path-connected spaces (otherwise, we basically care only about the path components of $$X$$ and $$Y$$). Consider the fact about::wedge sum:

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

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

Suppose, further that:


 * There exists an open subset $$A$$ of $$X$$ containing $$x_0$$ such that $$A$$ has a strong deformation retraction to $$x_0$$.
 * There exists an open subset $$B$$ of $$Y$$ containing $$y_0$$ such that $$B$$ has a strong deformation retraction to $$y_0$$.

Then, we have the following relationship between the fundamental groups of $$(X,x_0)$$, $$(Y,y_0)$$, and $$(Z,z_0)$$:

$$\! \pi_1(Z,z_0) \cong \pi_1(X,x_0) * \pi_1(Y,y_0)$$

where $$\! * $$ denotes the free product of groups.

Similar facts

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

Facts used

 * 1) uses::Seifert-van Kampen theorem