Hausdorff space

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Add Example:The real line with the finite complement topology is not Hausdorff as an example of a familiar set with an unfamiliar topology
Huge overhaul - removed utter nonsense Alec (talk) 22:49, 22 February 2017 (UTC)

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Definition

Given a topological space $(X,\mathcal{J})$ we say it is Hausdorff^[1] or satisfies the Hausdorff axiom if:

∀x1,x2∈X[x1≠x2⟹(∃N1∈N(x1,X)∃N2∈N(x2,X)[N1∩N2=∅])]^{[Note 1]}
- In words: for any two points in $X$ , if the points are distinct then there exist neighbourhoods to each point such that the neighbourhoods are disjoint
We may also write it: $\forall x_1,x_2\in X\exists N_1\in\mathcal{N}(x_1,X)\exists N_2\in\mathcal{N}(x_2,X)[x_1\neq x_2\implies N_1\cap N_2\eq\emptyset]$ ^{[Note 2]}^{[Note 3]}
It may also be said that in a Hausdorff space that "points may be separated by open sets"^[2]

A topological space satisfying this property is said to be a Hausdorff space^[2]

A Hausdorff space is sometimes called a T2 space

Equivalent definitions

∀x1,x2∈X[x1≠x2⟹(∃U1∈O(x1,X)∃U2∈O(x2,X)[U1∩U2=∅])]^[2] - see Claim 1
- In words: for all points in $X$ if the points are distinct then there exists open sets acting as open neighbourhoods to each point such that these open neighbourhoods are disjoint
- This, along the same thinking as for the definition, may be (and is commonly seen as) written: $\forall x_1,x_2\in X\exists U_1\in\mathcal{O}(x_1,X)\exists U_2\in\mathcal{O}(x_2,X)[x_1\neq x_2\implies U_1\cap U_2\eq\emptyset]$

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Proof of claims

Grade: D

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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Easy to prove the first and only claim on this page.

$\impliedby$ is easily seen as open sets are neighbourhoods (see "an open set is a neighbourhood to all of its points", the other way requires:

If $C\subseteq A$ and $D\subseteq B$ then if $A\cap B\eq\emptyset$ we have $C\cap D\eq\emptyset$ - this could be worth factoring out

but is otherwise really easy

This proof has been marked as an page requiring an easy proof

Further work for this page

Link to a theorem about all metric spaces being Hausdorff.
Proof of the equivalent form claims - I did say "no proof will be hand-waved away as trivial" but it certainly isn't worth my time now Alec (talk) 22:49, 22 February 2017 (UTC)

Notes

Jump up ↑ Note that if $X$ is the empty set, then there are no $x_1,x_2\in X$ , so the statement is vacuously true.
Jump up ↑ Again note that if $X$ is the empty set, then there are no $x_1,x_2\in X$ , so the statement is vacuously true. In the event $X$ has one or more points notice that then $X$ itself is an open set and an open set is a neighbourhood to all of its points, so there exists neighbourhoods, if we have points. Note lastly that if $x_1\eq x_2$ then we can pick this neighbourhood ( $X$ itself) and be done, as by the nature of logical implication we do not care about the truth or falsity of the $N_1\cap N_2\eq\emptyset$ part.
Jump up ↑ These are easily seen to be equivalent, try it! You need to do the $X$ is one point case, $X$ is empty and then $X$ contains 2 or more points - this is the easiest

References

[2] Jump up ↑ Note that if $X$ is the empty set, then there are no $x_1,x_2\in X$ , so the statement is vacuously true.

[3] Jump up ↑ Again note that if $X$ is the empty set, then there are no $x_1,x_2\in X$ , so the statement is vacuously true. In the event $X$ has one or more points notice that then $X$ itself is an open set and an open set is a neighbourhood to all of its points, so there exists neighbourhoods, if we have points. Note lastly that if $x_1\eq x_2$ then we can pick this neighbourhood ( $X$ itself) and be done, as by the nature of logical implication we do not care about the truth or falsity of the $N_1\cap N_2\eq\emptyset$ part.

[4] Jump up ↑ These are easily seen to be equivalent, try it! You need to do the $X$ is one point case, $X$ is empty and then $X$ contains 2 or more points - this is the easiest

[ITTBM-1] Jump up ↑ Introduction to Topology - Bert Mendelson

[ITTMJML-5] {Jump up to: 2.0} ^2.1 ^2.2 Introduction to Topological Manifolds - John M. Lee

[1]

[Note 1]

[Note 2]

[Note 3]

[2]

Hausdorff space

Contents

Definition

Equivalent definitions

See next

Proof of claims

Further work for this page

Notes

References

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