Passing to the quotient (topology)/Statement

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Statement

f descends to the quotient

Suppose that (X,J) is a topological space and is an equivalence relation, let (X,Q) be the resulting quotient topology and π:XX the resulting quotient map, then:

  • Let (Y,K) be any topological space and let f:XY be a continuous map that is constant on the fibres of π[Note 1] then:
  • there exists a unique continuous map, ˉf:XY such that f=¯fπ

We may then say f descends to the quotient or passes to the quotient

Note: this is an instance of passing-to-the-quotient for functions

Notes

  1. Jump up That means that:

References