I'm finding some problems in solving this exercise: let $X=[0,1] \times [0,1]$ and $A= \left \{ 0 \right \} \times [0,1]$.
- Show that $X/A=\left ([0,1] \times [0,1] \right )/ \left ( \left \{ 0 \right \} \times [0,1] \right )$ is homeomorphic to a triangle.
- Show that $X/A$ is homeomorphic to $X$.
Regarding point 1, my goal is to answer without using compactness, writing a continuos and bijective function $f: X \rightarrow Y$, where $Y$ is a triangle, constant on the equivalente classes of $X/A$. This implies that the function $[f]: X/A \rightarrow Y$ is continuos and bijective too. Since the composition $f= [f] \circ \pi$ is commutative (where $\pi$ is the natural projection) I'll have to show that also $f^{-1}$ is continuos, and then $[f]^{-1}$ is continuos too. Hence $[f]$ is an homeomorphism.
How to formalise all this? I need a general method, so that I can solve point 2 by myself.
Hint : Try to prove that the function $g : X \rightarrow X$ defined by $$g(x,y)=(x,xy)$$
defines a injection $\tilde{g} : X/A \rightarrow X$, and show that $\tilde{g}$ is a homeomorphism between $X/A$ and the triangle of vertices $(0,0)$, $(1,0)$ and $(1,1)$.