Quotient space (topology): Difference between revisions

Given an [[equivalence relation]] <math>\,\sim\,</math> on <math>X,</math> the canonical map <math>q : X \to X / {\sim}</math> that sends <math>x \in X</math> to its [[equivalence class]] <math>[x] := \{ z \in X : z \sim x \}</math> (that is, <math>q(x) := [x]</math>) is a quotient map that satisfies <math>q(x) = q^{-1}(q(x))</math> for all <math>x \in X</math>; moreover, for all <math>a, b \in X,</math> <math>a \,\sim\, b</math> if and only if <math>q(a) = q(b).</math>

In fact, let <math>\pi : X \to Y</math> be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all <math>a, b \in X</math> that <math>a \,\sim\, b</math> if and only if <math>\pi(a) = \pi(b).</math> Then <math>\,\sim\,</math> is an equivalence relation on <math>X</math> such that for every <math>x \in X,</math> <math>[x] = \pi^{-1}(\pi(x))</math> so that <math>\pi([x]) = \{ \,\pi(x)\, \} \subseteq Y</math> is a [[singleton set]], which thus induces a [[bijection]] <math>\hat{\pi} : X / {\sim} \;\;\to\; Y</math> defined by <math>\hat{\pi}([x]) := \pi(x)</math> (this is well defined because <math>\pi([x])</math> is a singleton set and <math>\hat{\pi}([x])</math> is just its unique element; that is, <math>\pi([x]) = \{ \,\hat{\pi}([x])\, \}</math> for every <math>x</math>).

Define the map <math>q : X \to X / {\sim}</math> as above (by <math>q(x) := [x]</math>) and give <math>X / \sim</math> the quotient topology induced by <math>q</math> (which makes <math>q</math> a quotient map). These maps are related by: <math display=block>\pi = \hat{\pi} \circ q \quad \text{ and } \quad q = \hat{\pi}^{-1} \circ \pi.</math>