Let $X=\mathbb{R} \sqcup \mathbb{R}$ and $\tau_x$ be the disjoint union topology. If $a\sim b \iff a\neq0\neq b \,\wedge\, a=b$, I would like to show that $Y=X/\sim$ is whether or not homeomorphic to $\mathbb{R}-\{0\}$.
I know that $\tau_Y$ is such that a set $V$ in $Y$ is open if $\pi^{-1}(V)$ is open in $X$ and $\pi$ is the canonical surjection such that $\pi(x)=[x]$. I know that if I find a continuous function $g$ such that $g(x)=g(x')$ whenever $x\sim x'$, then there exists an unique continuous function $f:Y\to \mathbb{R}-\{0\}$ such that $f=g\,\circ\,\pi$. Also I need to prove that the inverse map $f^{-1}$ is continuos to prove the homeomorphism.
How is $g$ defined?
Can somebody give a hand?
Any neighbourhoods of the two $0$'s in $Y$ will intersect non-trivially. Thus $Y$ is not Hausdorff. However $\mathbb{R}-\{0\}$ is Hausdorff. Thus they are not homeomorphic.