Difference between revisions of "Quotient topology/Equivalence relation definition"
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(Created page with "<noinclude> {{Requires references|grade=A|msg=See the notes page, the books are plentiful I just don't have them to hand.}} ==Definition== </noinclude>Given a topological sp...") |
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{{Requires references|grade=A|msg=See the notes page, the books are plentiful I just don't have them to hand.}} | {{Requires references|grade=A|msg=See the notes page, the books are plentiful I just don't have them to hand.}} | ||
==Definition== | ==Definition== | ||
| − | </noinclude>Given a [[topological space]], {{M|(X,\mathcal{J})}} and an [[equivalence relation]] on {{M|X}}, {{M|\sim}}, the ''quotient topology'' on {{M|\frac{X}{\sim} }}, {{M|\mathcal{K} }} is defined as: | + | </noinclude>Given a [[topological space]], {{M|(X,\mathcal{J})}} and an [[equivalence relation]] on {{M|X}}, {{M|\sim}}<ref group="Note"><!-- |
| + | |||
| + | -->Recall that for an [[equivalence relation]] there is a [[natural map]] that sends each {{M|x\in X}} to {{M|[x]}} (the [[equivalence class|equivalence class containing {{M|x}}]]) which we denote here as {{M|\pi:X\rightarrow\frac{X}{\sim} }}. | ||
| + | Recall also that {{M|\frac{X}{\sim} }} denotes the [[set of all equivalence classes|set of all equivalence classes of {{M|\sim}}]].<!-- | ||
| + | |||
| + | --></ref>, the ''quotient topology'' on {{M|\frac{X}{\sim} }}, {{M|\mathcal{K} }} is defined as: | ||
* The set {{M|\mathcal{K}\subseteq\mathcal{P}(\frac{X}{\sim})}} such that: | * The set {{M|\mathcal{K}\subseteq\mathcal{P}(\frac{X}{\sim})}} such that: | ||
** {{M|\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}]}} or equivalently | ** {{M|\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}]}} or equivalently | ||
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In words: | In words: | ||
* The topology on {{M|\frac{X}{\sim} }} consists of all those sets whose [[pre-image]] (under {{M|\pi}}) are [[open set|open]] in {{M|X}}<noinclude> | * The topology on {{M|\frac{X}{\sim} }} consists of all those sets whose [[pre-image]] (under {{M|\pi}}) are [[open set|open]] in {{M|X}}<noinclude> | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
</noinclude> | </noinclude> | ||
Latest revision as of 14:36, 25 April 2016
Grade: A
This page requires references, it is on a to-do list for being expanded with them.
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See the notes page, the books are plentiful I just don't have them to hand.
Definition
Given a topological space, [ilmath](X,\mathcal{J})[/ilmath] and an equivalence relation on [ilmath]X[/ilmath], [ilmath]\sim[/ilmath][Note 1], the quotient topology on [ilmath]\frac{X}{\sim} [/ilmath], [ilmath]\mathcal{K} [/ilmath] is defined as:
- The set [ilmath]\mathcal{K}\subseteq\mathcal{P}(\frac{X}{\sim})[/ilmath] such that:
- [ilmath]\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}][/ilmath] or equivalently
- [ilmath]\mathcal{K}=\{U\in\mathcal{P}(\frac{X}{\sim})\ \vert\ \pi^{-1}(U)\in\mathcal{J}\}[/ilmath]
In words:
- The topology on [ilmath]\frac{X}{\sim} [/ilmath] consists of all those sets whose pre-image (under [ilmath]\pi[/ilmath]) are open in [ilmath]X[/ilmath]
Notes
- ↑ Recall that for an equivalence relation there is a natural map that sends each [ilmath]x\in X[/ilmath] to [ilmath][x][/ilmath] (the equivalence class containing [ilmath]x[/ilmath]) which we denote here as [ilmath]\pi:X\rightarrow\frac{X}{\sim} [/ilmath]. Recall also that [ilmath]\frac{X}{\sim} [/ilmath] denotes the set of all equivalence classes of [ilmath]\sim[/ilmath].
References
