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
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| [ilmath]f[/ilmath] descends to the quotient |
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Suppose that [ilmath](X,\mathcal{ J })[/ilmath] is a topological space and [ilmath]\sim[/ilmath] is an equivalence relation, let [ilmath](\frac{X}{\sim},\mathcal{ Q })[/ilmath] be the resulting quotient topology and [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath] the resulting quotient map, then:
- Let [ilmath](Y,\mathcal{ K })[/ilmath] be any topological space and let [ilmath]f:X\rightarrow Y[/ilmath] be a continuous map that is constant on the fibres of [ilmath]\pi[/ilmath][Note 1] then:
- there exists a unique continuous map, [ilmath]\bar{f}:\frac{X}{\sim}\rightarrow Y[/ilmath] such that [ilmath]f=\overline{f}\circ\pi[/ilmath]
Notes
- ↑
That means that:
- [ilmath]\pi(x)=\pi(y)\implies f(x)=f(y)[/ilmath] - exactly as in quotient (function)
References
