My exercise is:
Let $F_n$ be the free group on $n$ generators. Determine an infinite succession of topological spaces $X_1$, $X_2$, $\ldots$ connected by arcs and such that the following properties are valid:
- $X_i$ is not homeomorphic to $X_j$ for $i \neq j$;
- the fundamental group of $X_i$ is equal to $F_n$ for every $i$ and for every $n$.
I plan to consider $X_i={}$ ($S_1 \wedge S_1 \wedge \ldots$ infinite times) united with infinite points. For example, $X_1=(S_1) \cup$ one point, $X_2=(S_1 \wedge S_1) \cup$ two points, et cetera. This space that I considered, is it connected by arcs? And does this space satisfy those two properties?