TITLE: Example of quotient mapping that is not open
QUESTION [24 upvotes]: I have the following definition: 
Let ($X$,$\mathcal{T}$) and ($X'$, $\mathcal{T'}$) be topological spaces. A surjection $q: X \longrightarrow X'$ is a quotient mapping if $$U'\in \mathcal{T'} \Longleftrightarrow q^{-1}\left( U'\right) \in \mathcal{T} \quad \text{i.e. if } \mathcal{T'}=\{ U' \subset X' : q^{-1}\left( U' \right) \in \mathcal{T} \}$$
and the properties:

$q$ is a bijective quotient mapping $\Leftrightarrow$ $q$ is a homeomorphism 
In general, $q$ quotient $\not \Rightarrow q$ open. 
If $U \in \mathcal{T}, q(U)\subset X'$ is open if $q^{-1}\left( q\left( U \right) \right) \in \mathcal{T}$ but not in general.

I could not find an example of quotient mapping for which $q^{-1}\left( q\left( U \right) \right)$ is not open. I would understand the idea better if you could show me one.

REPLY [6 votes]: Here is a "visual" example: 
Wrap the unit interval $[0,1]$ around the unit circle $S^1$ by identifying $0$ and $1$. The interval $[0,1/2) $ is open in $[0,1]$, but its image in $S^1$ is not open.