# Expressing $\mathbb{R}$ as the quotient of a disjoint union of unit intervals

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.

• Just an idea (maybe, too trivial) - what about starting with $i=0$ interval being $[0,1]\subset \Bbb R$ and attaching $2i$-th interval to the left and $2i+1$-th interval to the right? – Ilya Aug 22 '12 at 15:50
• I thought about that, but I was trying to find a way that would make make the identifications easier. Maybe I should go back to this one. – Holdsworth88 Aug 22 '12 at 15:51
• I see. I think Brian's idea of using integers is better as it allows avoiding such mess. – Ilya Aug 22 '12 at 15:56

Your idea is fine with just a small modification: $\Bbb R=\amalg_{n\in A}I_n/(1_n\sim 0_{n+1})$. To make life easier, you should also use $A$ as the index set for $\amalg_n S_n^1/P$, where $P$ is the set of distinguished points.