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==Definition==
==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>.<ref>C. Adams and R. Franzosa. ''Introduction to Topology: Pure and Applied''. Upper Saddle River, NJ: Pearson Prentice Hall, 2008. p. 89</ref>
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>.<ref>C. Adams and R. Franzosa. ''Introduction to Topology: Pure and Applied''. Upper Saddle River, NJ: Pearson Prentice Hall, 2008. p. 89</ref>
==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) = 1</math> and <math>p(x) = x</math> for <math>x \neq 1</math>. Then <math>A</math> under the quotient topology is [[homeomorphism|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.
==References==
==References==