Exercise 6, § 5

Let $X$ be the numerical right $\mathbf{R}$ and let $A$ be a subset of $X$. Let $Y$ be the space obtained from $X$ by contracting $A$ to a point.

a) If $A$ is discrete, the Poincaré group of $Y$ is a free group.

b) If $A$ has an accumulation point, the Poincaré group of $Y$ has the power of the continuum and is not free.

c) If $A$ is the closure of the set of points $1/n$, for $n\in \mathbf{N}^*$, the space $X/A$ is homeomorphic to the Hawaiian earring.