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Quotient Space -- from Wolfram MathWorld

️Weisstein, Eric W.

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The quotient space X/∼ of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of X/∼ : a subset of X/∼ is called open iff union _([a] in U)a is open in . 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 -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 -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, is interpreted as the space obtained when the boundary of the -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."