Definition:Quotient Topology/Quotient Space

Definition
Given a space $$X \ $$ with topology $$\vartheta$$ and an equivalence relation $$\sim \ $$ on $$X \ $$, the quotient space $$X/ \sim \ $$ is defined as the space whose points are equivalence classes of $$\sim \ $$, (called the quotient set and whose topology $$\vartheta_{X/ \sim}$$ is defined as:


 * $$U \in \vartheta_{X/ \sim} \iff \pi^{-1}(U) \in \vartheta \ $$

where $$\pi:X \to X/ \sim \ $$ is the function taking a point in $$X \ $$ to its equivalence class, called the quotient mapping.