Difference between revisions of "Hausdorff space"
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(Created page with "==Definition== Given a Topological space {{M|(X,\mathcal{J})}} we say it is '''Hausdorff'''<ref>Introduction to topology - Mendelson - Third Edition</ref> or '''satisfies...") |
(Making reference a book-reference, adding alternative definition, marking stub, adding outline of proof definitions are the same. Added notes about further work required on page) |
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| + | {{Refactor notice|grade=A|msg=Page was 1 year and 1 day since modification, basically a stub, seriously needs an update.}} | ||
==Definition== | ==Definition== | ||
| − | Given a [[Topological space]] {{M|(X,\mathcal{J})}} we say it is '''Hausdorff''' | + | Given a [[Topological space]] {{M|(X,\mathcal{J})}} we say it is '''Hausdorff'''{{rITTBM}} or '''satisfies the Hausdorff axiom''' if: |
* For all {{M|a,b\in X}} that are distinct there exists [[Open set#Neighbourhood 2|neighbourhoods]] to {{M|a}} and {{M|b}}, {{M|N_a}} and {{M|N_b}} such that: | * For all {{M|a,b\in X}} that are distinct there exists [[Open set#Neighbourhood 2|neighbourhoods]] to {{M|a}} and {{M|b}}, {{M|N_a}} and {{M|N_b}} such that: | ||
** {{M|1=N_a\cap N_b=\emptyset}} | ** {{M|1=N_a\cap N_b=\emptyset}} | ||
| + | ===Alternate definition=== | ||
| + | * {{M|1=\forall a,b\in X\exists A,B\in\mathcal{J}[a\ne b\implies A\cap B=\emptyset]}}{{rITTMJML}} | ||
| + | {{Requires proof|Are these statements the same? Clearly {{M|\text{neighbourhood }\implies\text{open-set} }} as a neighbourhood to a point requires the existence of an open set containing that point (contained in the neighbourhood) and clearly {{M|\text{open-set}\implies\text{neighbourhood} }} as an open set ''is'' a neighbourhood - write this up.}} | ||
| + | ==Further work for this page== | ||
| + | * Link to a theorem about all metric spaces being Hausdorff. | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | + | {{Topology navbox|plain}} | |
{{Definition|Topology}} | {{Definition|Topology}} | ||
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Definition
Given a Topological space [ilmath](X,\mathcal{J})[/ilmath] we say it is Hausdorff[1] or satisfies the Hausdorff axiom if:
- For all [ilmath]a,b\in X[/ilmath] that are distinct there exists neighbourhoods to [ilmath]a[/ilmath] and [ilmath]b[/ilmath], [ilmath]N_a[/ilmath] and [ilmath]N_b[/ilmath] such that:
- [ilmath]N_a\cap N_b=\emptyset[/ilmath]
Alternate definition
- [ilmath]\forall a,b\in X\exists A,B\in\mathcal{J}[a\ne b\implies A\cap B=\emptyset][/ilmath][2]
(Unknown grade)
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
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The message provided is:
The message provided is:
Are these statements the same? Clearly [ilmath]\text{neighbourhood }\implies\text{open-set} [/ilmath] as a neighbourhood to a point requires the existence of an open set containing that point (contained in the neighbourhood) and clearly [ilmath]\text{open-set}\implies\text{neighbourhood} [/ilmath] as an open set is a neighbourhood - write this up.
Further work for this page
- Link to a theorem about all metric spaces being Hausdorff.
References
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