Example of quotient mapping that is not open 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. 

An open map is:
for any open set in the domain, its image is open
A quotient map only requires:
for any open preimage in the domain, its image is open
A quotient map may not be an open map because there may exist a set such that it is open but is not a preimage, i.e., the preimage of the image of an open set may not be the open set itself.
