Quotient topology

Quotient topology is a concept in the branch of mathematics known as topology.

Definition

Let <math>X</math> be a topological space, and <math>A</math> a set, and let <math>p \colon X \to A</math> be a surjection. The quotient topology on <math>A</math> induced by <math>p</math> is the topology whose open sets are the sets <math>U \subseteq A</math> such that <math>p^{-1}(U)</math> is an open set in <math>X</math>.

Examples

Quotient topologies can often be visualized as gluing elements of a topological space together.

Let <math>X = [0, 1]</math> with the usual topology (as a subspace of the reals), <math>A = [0, 1)</math>, and <math>p \colon X \to A</math> be given by <math>p(1) = 0</math> and <math>p(x) = x</math> for <math>x \neq 1</math>. Then <math>A</math> under the quotient topology is homeomorphic to the circle. Indeed, we can visualize what happened as a gluing operation: the two endpoints of the interval were glued together to create a closed loop.