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Concept
Quotient space (topology)
Summary
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 under the canonical projection map is open in the original topological space.
Intuitively speaking, the points of each equivalence class are 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.
Let be a topological space, and let be an equivalence relation on The quotient set is the set of equivalence classes of elements of The equivalence class of is denoted
The construction of defines a canonical surjection As discussed below, is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated to
The quotient space under is the set equipped with the quotient topology, whose open sets are those subsets whose is open. In other words, is open in the quotient topology on if and only if is open in Similarly, a subset is closed if and only if is closed in
The quotient topology is the final topology on the quotient set, with respect to the map
A map is a quotient map (sometimes called an identification map) if it is surjective and is equipped with the final topology induced by The latter condition admits two more-elementary phrasings: a subset is open (closed) if and only if is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.
Official source
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