Quotient Topology

From Conservapedia

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

Definition[edit]

Let $\text{[math]}$ be a topological space, and $\text{[math]}$ a set, and let $\text{[math]}$ be a surjection. The quotient topology on $\text{[math]}$ induced by $\text{[math]}$ is the topology whose open sets are the sets $\text{[math]}$ such that $\text{[math]}$ is an open set in $\text{[math]}$ .^[1]

Examples[edit]

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

Let $\text{[math]}$ with the usual topology (as a subspace of the reals), $\text{[math]}$ , and $\text{[math]}$ be given by $\text{[math]}$ and $\text{[math]}$ for $\text{[math]}$ . Then $\text{[math]}$ 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.

References[edit]

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

Categories: [Topology]

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