$\mathcal{E}_2$ is the euclidean topology. We have an equivalent relation on $(\mathbb{R}^2,\mathcal{E}_2)$ with (x,y) $\mathcal{R}$ (x',y') $\Leftrightarrow$ x' = x and y'=$\pm y$.
I want to find whether or not the space $(\mathbb{R}^2,\mathcal{E}_2)$ with the quotient topology is compact or not. I think that it is compact because I then need to find an homeomorphism to the space H = $\{(x,y) \in \mathbb{R}^2 \vert y \geq 0\}$ with the induced topology on H from euclidean so the space is (H,$\mathcal{E}_H)$.
I think that H is compact with the induced topology, so I want to say that the quotient space is compact as well. If I can find that the quotient space is compact then they would be homeomorphic with f(x,y) = (x,$\vert y \vert$) because it is continuous and bijective from a compact space to Hausdorff space.
Any hints?