Quotient Space

$X/\sim$ is the set of equivalence relations on the topological spaces $X$ defined by the relation $\sim$.

Examples:

$X=\mathbb{R},a\sim b⟺a-b\in \mathbb{N}\Rightarrow X/\sim \approx S^1$ ( a circle ) $X={\mathbb{R}}^2,a\sim b⟺a-b\in \mathbb{R}\times \{0\}\Rightarrow X/\sim \approx \mathbb{R}$ ( a line ) $X={\mathbb{R}}^2,a\sim b⟺a-b\in \mathbb{N}\times \mathbb{N}\Rightarrow X/\sim \approx \mathbb{T}$. ( a torus )