Show $X+_fY$ is homeomorphic to $\mathbb{S}^1$. 
Know result: If $X$ is any space, $A$ is a subset of $X$, and $p\notin X$, the space $X+_f\{p\}$ resulting from the function $f$ which takes $A$ to $\{p\}$ is homeomorphic to the quotient space of $X$ obtained by identifying $A$ to a single point.


Problem: Let $X=[0,1]$, $Y=[2,3]$, $A=\{0,1\}$, and let $f\colon A\to Y$ be defined by $f(0)=2$, $f(1)=3$. Then $X+_fY$ is homeomorphic to $\mathbb{S}^1$.

I've finished the section over quotient topologies, in Willards text, and came across this problem. Willard calls such constructions "attachings". But, I really am not sure do such a problem. Any help is appreciated.
 A: The equivalence relation that gives you the quotient space $X+_fY$ has a one-element class for each $x\in(0,1)\cup(2,3)$ and two two-element classes, $\{0,2\}$ and $\{1,3\}$. The most straightforward homeomorphism would map $\{0,2\}$ to $\langle 1,0\rangle$, $\{1,3\}$ to $\langle 1,\pi\rangle$, the class $\{x\}$ for $x\in(0,1)$ to $\langle 1,\pi x\rangle$, and the class $\{x\}$ to $$\langle 1,\pi+(x-2)\pi\rangle=\langle 1,(x-1)\pi\rangle$$ for $x\in(2,3)$. (Here I’m giving the points of $\Bbb S^1$ in polar coordinates.)
In other words, the map wraps $(0,1)$ around the open upper semicircle of $\Bbb S^1$ and wraps $(2,3)$ around the open lower semicircle of $\Bbb S^1$
A: This appears to be an obtuse way to specify a quotient.  Take $\sim \,\subseteq X \times Y$ to be the (partial) relation $0 \sim 2$, $1 \sim 3$ and construct the quotient $(X \cup Y)/ \sim$.
Graphically, this is two line segments with left endpoints identified and right endpoints identified.
If you wish to proving all the way down to the axioms, identify $X$ with the closed unit upper semicircle and identify $Y$ with the closed unit lower semicircle by homeomorphisms.
\begin{align*}
f:X \rightarrow \Bbb{R}^2 &: x \mapsto (\cos \pi x, \sin \pi x)  \\
g:Y \rightarrow \Bbb{R}^2 &: y \mapsto (\cos \pi(y-2), -\sin \pi(y-2))
\end{align*}
Verify that the attaching identifies the two points that have been mapped to $(-1,0)$ and $(0,1)$ by the two homeomorphisms.  Verify that open sets in the unit circle are taken to open sets in the attachment by these two homeomorphisms quotient the attaching identifications and vice versa.
