Passing to the quotient (topology)/Statement
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Statement
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f descends to the quotient |
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Suppose that (X,J) is a topological space and ∼ is an equivalence relation, let (X∼,Q) be the resulting quotient topology and π:X→X∼ the resulting quotient map, then:
- Let (Y,K) be any topological space and let f:X→Y be a continuous map that is constant on the fibres of π[Note 1] then:
- there exists a unique continuous map, ˉf:X∼→Y such that f=¯f∘π
Notes
- Jump up ↑
That means that:
- π(x)=π(y)⟹f(x)=f(y) - exactly as in quotient (function)
References