TITLE: Quotient space definition
QUESTION [6 upvotes]: I am trying to solve a homework problem where a quotient space is defined in a way that I do not understand, so I will ask a more simplified question.
I understand that $$ [0,1]/0  \sim  1, $$
means that you sew the points 0 and 1 together, getting a circle. The equivalence classes are all singletons except for the one that contains 0 and 1. There is also the common example where a ball is converted into a sphere by sewing the boundary of the ball together.
But what does the quotient space defined by
$$ [-1, 1]/[0,1/2]$$
look like? The notation is telling me that all points $[0,1/2]$ are identified to a single point, but I still don't understand what this operation means exactly. 
I drew a sketch of what I thought happened when $[0,1/2]$ was identified to a single point:

I would really like to understand what this space looks like, and what "identified to a single point" means for this problem.

REPLY [6 votes]: It’s much simpler than your picture: it just contracts the interval $\left[0,\frac12\right]$ to a single point. The resulting space is homeomorphic to a closed interval of $\Bbb R$. Let $p$ be the point of the quotient corresponding to the interval $\left[0,\frac12\right]$. Perhaps the most obvious homeomorphism is
$$h:[-1,1]/[0,1/2]\to[-1,1/2]:x\mapsto\begin{cases}
x,&\text{if }-1\le x<0\\\\
0,&\text{if }x=p\\\\
x-\frac12,&\text{if }\frac12<x\le 1\;.
\end{cases}$$