Quotient Space

The quotient space X/∼ 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/∼: a subset U of X/∼ is called open iff  union _([a] in U)a is open in X. Quotient spaces are also called factor spaces.

This can be stated in terms of maps as follows: if q:X->X/∼ denotes the map that sends each point to its equivalence class in X/∼, the topology on X/∼ can be specified by prescribing that a subset of X/∼ is open iff q^(-1)[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/∼->Y is continuous iff the function f degreesq:X->Y is continuous.

Let D^n be the closed n-dimensional disk and S^(n-1) its boundary, the (n-1)-dimensional sphere. Then D^n/S^(n-1) (which is homeomorphic to S^n), provides an example of a quotient space. Here, D^n/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 S^(n-1) are equivalent."

See also

Equivalence Relation, Lie Group Quotient Space, Topological Space

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Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

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Quotient Space

Cite this as:

Weisstein, Eric W. "Quotient Space." From MathWorld--A Wolfram Web Resource.

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