Difference between revisions of "Passing to the quotient (topology)/Statement"
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==Statement== | ==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== | ||
Latest revision as of 20:23, 11 October 2016
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Statement
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
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
- Jump up ↑
That means that:
- ∀x,y∈X[π(x)=π(y)⟹f(x)=f(y)] - as mentioned in passing-to-the-quotient for functions, or
- ∀x,y∈X[f(x)≠f(y)⟹π(x)≠π(y)], also mentioned
- See :- Equivalent conditions to being constant on the fibres of a map for details
References
