I am trying to complete exercise 3.18(a) in Lee's Introduction to Topological Manifolds. The exercise is as follows:
Let $A \subseteq \mathbb{R}$ be the set of integers and let $X$ be the quotient space $\mathbb{R}/A$, where $A$ is collapsed down to a point.
(a) Show that $X$ is homeomorphic to a wedge sum of countably infinitely many circles.
Lee gives the hint that one should express both spaces as quotients of a disjoint union of intervals, implying that one will make use of the uniqueness of quotient spaces.
The wedge sum of countably infinitely many circles is the space $B=\bigvee_{i=0} ^\infty S_i ^1=\amalg_{i=0} ^\infty S_i ^1/\{p_i\}$, where $p_i \in S_i ^1$ and $\{p_i\}$ denotes the relation $p_i\sim p_j$ for all $i,j$. Since $S^1$ is the quotient space $I/(0\sim1)$, we can express $B$ as $B=\amalg_{i=0} ^\infty I_i/\sim$, where $\sim$ is the relationship $p_i \sim p_j$ for all $i,j$ and $0_i\sim 1_i$ for all $i$. With this one defines the quotient map $q:\amalg_{i=0} ^\infty I_i \rightarrow B$ by sending elements to their equivalence classes.
I am stuck trying come up with a similar formulation for $\mathbb{R}$. Graphically I could imagine $\mathbb{R}$ being constructed by attaching countably infinitely many of the unit intervals $I_i$ together with the identification $1_i\sim 0_{i+1}$. Formally this would be $\mathbb{R}=\amalg_{i=0} ^\infty I_i/(1_i\sim 0_{i+1})$. Using this idea though I can only see myself being able to construct the positive real numbers $\mathbb{R}^+$ which is not homeomorphic to $\mathbb{R}$. Is their some way to save this idea?
If I could figure this out, then I would have to find a quotient map $r:\amalg_{i=0} ^\infty I_i\rightarrow \mathbb{R}/A$ that makes the same identifications as $q$ to prove that the two spaces are homeomorphic by the uniqueness of the quotient space.