# Quotient topology

### From Conservapedia

**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 be a surjection. The quotient topology on *A* induced by *p* is the topology whose open sets are the sets 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 be given by *p*(1) = 1 and *p*(*x*) = *x* for . 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

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