Example 4

Let $((X_i, x_i))_{i\in I}$ be a family of pointed topological spaces. The bouquet of the family $((X_i, x_i))_{i\in I}$ is by definition, and is denoted by $\bigvee_{i\in I}(X_i, x_i)$, the quotient topological space of the sum space of the family $(X_i)_{i\in I}$ by the equivalence relation which identifies all the points $(x_i, i)$ with one another, for $i\in I$. Let X denote this topological space and $x$ the common image of the $x_i$. Suppose that, for every $i\in$ I, the point $x_i$ is closed in $X_i$ and that the spaces $X_i$ are simply connected. If I is finite, Corollary 4 implies that the canonical homomorphism $_i*_{\in I}\pi_1(X_i, x_i)\rightarrow \pi_1(X, x)$ is an isomorphism.

Remark 1 of IV, p. 429 and Exercise 3, IV, p. 463 give less restrictive conditions under which this homomorphism is an isomorphism. See nevertheless Exercise 4, IV, p. 464.