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Let $q:\mathbb{R}^2\longrightarrow \mathbb{R}$ be the projection over the first coordinate, $(x,y)\longmapsto x$. Projections are always open, so $q$ is quotient map. So we have proved that when identifying the points with the same first coordinate in the plane, the resulting quotient space is homeomorphic to the real line. Note that $q$ is not closed (the hyperbola $\{xy=1\}$, which is closed, is projected onto $\mathbb{R}\smallsetminus \{0\}$, which is not closed).