Quotient topology

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

Definition

Let X be a topological space, and A a set, and let p \colon X \to A be a surjection. The quotient topology on A induced by p is the topology whose open sets are the sets U \subseteq A such that p − 1(U) is an open set in X.[1]

Examples

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

Let X = [0,1] with the usual topology (as a subspace of the reals), A = [0,1), and p \colon X \to A be given by p(1) = 1 and p(x) = x for x \neq 1. Then A 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.

References

  1. C. Adams and R. Franzosa. Introduction to Topology: Pure and Applied. Upper Saddle River, NJ: Pearson Prentice Hall, 2008. p. 89
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