Quotient topology: Difference between revisions
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In [[general topology]], the '''quotient topology''' is defined on the [[image]] of a [[topological space]] under a [[function (mathematics)|function]]. | In [[general topology]], the '''quotient topology''', or '''identification topology''' is defined on the [[image]] of a [[topological space]] under a [[function (mathematics)|function]]. | ||
Let <math>(X,\mathcal T)</math> be a topological space, and ''q'' a [[surjective function]] from ''X'' onto a set ''Y''. The quotient topology on ''Y'' has as open sets those subsets <math>U</math> of <math>Y</math> such that the [[pre-image]] <math>q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in \mathcal T_X</math>. The quotient topology has the [[universal property]] that it is the finest topology such that ''q'' is a [[continuous map]]. | Let <math>(X,\mathcal T)</math> be a topological space, and ''q'' a [[surjective function]] from ''X'' onto a set ''Y''. The quotient topology on ''Y'' has as open sets those subsets <math>U</math> of <math>Y</math> such that the [[pre-image]] <math>q^{-1}(U)=\{x \in X \mid q(x) \in U \} \in \mathcal T_X</math>. | ||
The quotient topology has the [[universal property]] that it is the finest topology such that ''q'' is a [[continuous map]]. | |||
==References== | |||
* {{cite book | author=Wolfgang Franz | title=General Topology | publisher=Harrap | year=1967 | pages=56 }} | |||
* {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=94-99 }} | |||
* {{cite book | author=Lynn Arthur Steen | authorlink=Lynn Arthur Steen | coauthors= J. Arthur Seebach jr | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 | pages=9 }} | |||
Revision as of 02:32, 31 December 2008
In general topology, the quotient topology, or identification topology is defined on the image of a topological space under a function.
Let be a topological space, and q a surjective function from X onto a set Y. The quotient topology on Y has as open sets those subsets of such that the pre-image .
The quotient topology has the universal property that it is the finest topology such that q is a continuous map.
References
- Wolfgang Franz (1967). General Topology. Harrap, 56.
- J.L. Kelley (1955). General topology. van Nostrand, 94-99.
- Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 9. ISBN 0-387-90312-7.