A quotient topology exercise.

Question

Define the quotient space:

(1) The real line $\mathbb{R}$ with $[-1,1]$ collapsed to a point.

(2) The real line $\mathbb{R}$ with $(-1,1)$ collapsed to a point.

The first one I think the quotient space would just be the real number line. I don't know about the second one. Would it be the same thing?

Answer 1

The second quotient space is just a weird topological space. Notice some oddities:

• The point $[(-1,1)]$ is open.
• Any open set containing $[1]$ also contains the point $[(-1,1)]$ (likewise for the point $[-1]$).

In particular, the space is not Hausdorff, and it is certainly not homeomorphic to $\mathbb{R}$. It is however connected, since it is the continuous image of a connected space.