Difference between revisions of "Passing to the quotient (topology)/Statement"
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That means that: | That means that: | ||
| − | * {{M|1=\pi(x)=\pi(y)\implies f(x)=f(y)}} - | + | * {{M|1=\forall x,y\in X[\pi(x)=\pi(y)\implies f(x)=f(y)]}} - as mentioned in [[passing to the quotient (function)|passing-to-the-quotient for functions]], or |
| + | * {{M|1=\forall x,y\in X[f(x)\ne f(y)\implies \pi(x)\ne\pi(y)]}}, also mentioned | ||
| + | * See :- [[Equivalent conditions to being constant on the fibres of a map]] for details | ||
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--></ref> then: | --></ref> then: | ||
* there exists a unique continuous map, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} such that {{M|1=f=\overline{f}\circ\pi }} | * there exists a unique continuous map, {{M|\bar{f}:\frac{X}{\sim}\rightarrow Y}} such that {{M|1=f=\overline{f}\circ\pi }} | ||
| − | We may then say {{M|f}} ''descends to the quotient'' or ''passes to the quotient''<div style="clear:both;"></div> | + | We may then say {{M|f}} ''descends to the quotient'' or ''passes to the quotient'' |
| + | : '''Note: ''' this is an instance of ''[[passing to the quotient (function)|passing-to-the-quotient for functions]]''<div style="clear:both;"></div> | ||
<noinclude> | <noinclude> | ||
==Notes== | ==Notes== | ||
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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]
We may then say [ilmath]f[/ilmath] descends to the quotient or passes to the quotient
- Note: this is an instance of passing-to-the-quotient for functions
Notes
- ↑
That means that:
- [ilmath]\forall x,y\in X[\pi(x)=\pi(y)\implies f(x)=f(y)][/ilmath] - as mentioned in passing-to-the-quotient for functions, or
- [ilmath]\forall x,y\in X[f(x)\ne f(y)\implies \pi(x)\ne\pi(y)][/ilmath], also mentioned
- See :- Equivalent conditions to being constant on the fibres of a map for details
References
