Quotient Space

The quotient space of a Topological Space and an Equivalence Relation ~ on is theset of Equivalence Classes of points in (under the Equivalence Relation )together with the following topology given to subsets of : a subset of is called open Iff is open in .

This can be stated in terms of Maps as follows: if denotes the Map that sends eachpoint to its Equivalence Class in , the topology on can be specified by prescribing that asubset of is open Iff is open.

In general, quotient spaces are not well behaved, and little is known about them. However, it is known that any compactmetrizable space is a quotient of the Cantor Set, any compact connected -dimensional Manifold for is aquotient of any other, and a function out of a quotient space is continuous Iff the function is continuous.

Let be the closed -D Disk and its boundary, the -D sphere. Then (which is homeomorphic to ), provides an example of a quotient space. Here, is interpreted as thespace obtained when the boundary of the -Disk is collapsed to a point, and is formally the ``quotient space by theequivalence relation generated by the relations that all points in are equivalent.''

References

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

• Schur Matrix
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