Quotient Space

The quotient space $X/\!\!\sim$ of a Topological Space $X$ and an Equivalence Relation ~ on $X$ is the set of Equivalence Classes of points in $X$ (under the Equivalence Relation ) together with the following topology given to subsets of $X/\!\!\sim$: a subset $U$ of $X/\!\!\sim$ is called open Iff $\cup_{[a]\in U} a$ is open in $X$.

This can be stated in terms of Maps as follows: if $q: X \to X /\!\!\sim$ denotes the Map that sends each point to its Equivalence Class in $X/\!\!\sim$, the topology on $X/\!\!\sim$ can be specified by prescribing that a subset of $X/\!\!\sim$ is open Iff $q^{-1}[{\rm the\ set}]$ is open.

In general, quotient spaces are not well behaved, and little is known about them. However, it is known that any compact metrizable space is a quotient of the Cantor Set, any compact connected $n$-dimensional Manifold for $n>0$ is a quotient of any other, and a function out of a quotient space $f: X/\!\!\sim \to Y$ is continuous Iff the function $f\circ q:X\to Y$ is continuous.

Let $\Bbb{D}^n$ be the closed $n$-D Disk and $\Bbb{S}^{n-1}$ its boundary, the $(n-1)$-D sphere. Then $\Bbb{D}^n/\Bbb{S}^{n-1}$ (which is homeomorphic to $\Bbb{S}^n$), provides an example of a quotient space. Here, $\Bbb{D}^n/\Bbb{S}^{n-1}$ is interpreted as the space obtained when the boundary of the $n$-Disk is collapsed to a point, and is formally the ``quotient space by the equivalence relation generated by the relations that all points in $\Bbb{S}^{n-1}$ are equivalent.''

See also Equivalence Relation, Topological Space

References

Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975.