Quotient Space


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What is Quotient space (topology)?

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.

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acteventgeneral topologygroup actiongroup actions (mathematicsquotient objecttopology


GluingIdentifiation mapIdentification mapIdentification spaceQuotient mapQuotient topological spaceQuotient topology


Espaço topológico quociente (pt)Quotiententopologie (de)Quotiënttopologie (nl)Topología cociente (es)Topologia ilorazowa (pl)Topologia quoziente (it)Topologie quotient (fr)Факторпростір (uk)Факторпространство (ru)몫공간 (ko)商位相空間 (ja)商空间 (zh)

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 — Date merged: 11/6/2021, 1:32:40 PM
 — Date scraped: 5/20/2021, 6:16:25 PM