Quotient Space
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Quotient spaces are also called factor spaces.
Quotient space. The quotient space 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. In topology and related areas of mathematics a quotient space is the quotient set of a topological space under an equivalence relation which is equipped with the quotient topology that is the finest topology the topology that has the largest set of open sets that makes continuous the canonical. If denotes the map that sends each point to its equivalence class in the topology on can be specified by prescribing that a subset of is.
The quotient space x s has as its elements all distinct cosets of x modulo s. This can be stated in terms of maps as follows. Ifv v then we denote by.
Definition let fbe a field va vector space over fandw va subspace ofv. 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. The space obtained is called a quotient space and is denoted v n read v mod n or v by n.
Forv1 v2 v we say thatv1 v2modwif and only ifv1 v2 w. Unfortunately a different choice of inner product can change. The quotient space is an abstract vector space not necessarily isomorphic to a subspace of.
With the natural definitions of addition and scalar multiplication x s is a linear space. A subset of is called open iff is open in.
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