Difference between revisions of "Passing to the quotient (topology)/Statement"

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Revision as of 13:15, 27 April 2016


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Statement

f descends to the quotient

Suppose that (X,\mathcal{ J }) is a topological space and \sim is an equivalence relation, let (\frac{X}{\sim},\mathcal{ Q }) be the resulting quotient topology and \pi:X\rightarrow\frac{X}{\sim} the resulting quotient map, then:

  • Let (Y,\mathcal{ K }) be any topological space and let f:X\rightarrow Y be a continuous map that is constant on the fibres of \pi[Note 1] then:
  • there exists a unique continuous map, \bar{f}:\frac{X}{\sim}\rightarrow Y such that f=\overline{f}\circ\pi
We may then say f descends to the quotient or passes to the quotient

Notes

  1. Jump up That means that:
    • \pi(x)=\pi(y)\implies f(x)=f(y) - exactly as in quotient (function)

References