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.