Hausdorff and separable spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 4 March 2014

Hausdorff spaces

Let X be a set. The diagonal map is

Δ: XX×X given by Δ(x) =(x,x) .
The image of the diagonal map Δ(X) is the graph of the identify function idX: XX.
A Hausdorff space is a topolgical space such that Δ(X) is closed in X×X, where X×X has the product topology.

Let X be a topological space. The following are equivalent.

(H) If x,yX and xy then there exist U𝒩(x) and V𝒩(y) such that UV=.
(HI) If xX then N𝒩(x)N={x}.
(HII) If Δ:XX×X is the diagonal map then Δ(X) is closed in X×X.
(HIII) If I is a set and Δ:XkIXk, where Xk=X for kI, is the diagonal map then Δ(X) is closed in kIXk.
(HIV) If 𝒢 is a filter on X then 𝒢 has at most one limit point.
(HV) If 𝒥 is a filter on X and x is a limit point of 𝒥 then x is the only cluster point of 𝒥.

Proof.

(HIII)(HII): HII is a special case of HIII.

(HII)(H): Assume x,yX and xy.
Then (x,y)X×X and (x,y)Δ(X).
Thus, by (HII), (x,y)Δ(X).
So (x,y) is not a close point of Δ(X).
So there exists a neighborhood Z𝒩((x,y)) such that ZΔ(X)=.
By the definition of the product topology, there exist U𝒩(x) and V𝒩(y) such that (U×V)Δ(X)=.
So UB=.

(H)(HIII): Assume (H).
To show: Δ(X) is closed in kIXk, where Xk=X.
To show: If xkIXk and xΔ(X) then x is not a close point of Δ(X).
Assume x=(xk)kIXk and xΔ(X).
To show: There exists W𝒩(x) such that WΔ(X)=.
Let i,jI be such that xixj.
Let Vi𝒩(xi) and Vj𝒩(xj) such that ViVj=.
Then W=Vi×Vj×ki,jXk𝒩(x) and WΔ(X)=.
So x is not a close point of Δ(X).
So Δ(X) is closed in kIXk.

(H)(HI): Assume (H).
To show: If xX then N𝒩(x)N={x}.
Assume xX.
To show: If yX and y{x} then yN𝒩(x)N.
Assume yX and y{x}.
To show: There exists U𝒩(x) such that yU.
By (H), since yx, there exist U𝒩(x) and V𝒩(y) such that UV=.
So yU.
So yN𝒩(x)N.

(HI)(HV): Assume (HI).
Assume 𝒥 is a filter on X and x is a limit point of 𝒥.
So 𝒥𝒩(x).
Let yX with yx.
To show: y is not a cluster point of 𝒥.
To show: yM𝒥M.
To show: There exists M𝒥 such that yM.
By (HI), yN𝒩(x)N.
So there exists M𝒩(x) such that yM.
Since 𝒥𝒩(x), then M𝒥.
So y is not a cluster point of 𝒥.

(HV)(HIV): Assume (HV).
Let 𝒢 be a filter on X and let x be a limit point of 𝒢.
Let yX with yx.
To show: y is not a limit point of 𝒢.
If y is a limit point of 𝒢 then y is a cluster point of 𝒢, which is a contradiction to (HV).
So y is not a limit point of 𝒢.

(HIV)(H): Assume not (H).
Let x,yX with xy such that there do not exist U𝒩(x) and V𝒩(y) with UV=.
Let 𝒥 be the filter generated by { UV|U 𝒩(x),V 𝒩(y) } . Then x and y are both limit points of 𝒥.
So (HIV) does not hold.

HW: Show that metric spaces are Hausdorff.

Separable spaces

Remove separability, this must have to do with normed linear spaces. See [Bou, Top, Ch. 1 Sec 1 Ex 7].

Separability appears in [BR] Chapter 2 Exercises 22 and 23, and in [Ru] Chapter 4 Exercises 2, 3, 4 and 18. A uniform space is almost a metric space: By [Bou??] the separable Hausdorff uniform spaces are exactly the separable metric spaces.

A topological space X is separable if it has a countable base, or?? if it has a countable dense set.

Hausdorff separable spaces are separable uniform spaces and metric spaces???

HW: Give an example of a metric space that is not separable. See [Ru] Chapter 4 Exercises 3 and 18. The key examples are p is separable if p and is not separable.

HW: Show that a Hilbert space is separable if and only if it contains a maximal orthonormal system which is at most countable. (See [Ru, Ch. 4 Ex. 4]).

HW: Give an example of a uniform space that is not separable.

HW: Show that k is separable.

Notes and References

These notes follow Bourbaki [Bou, Ch.I §8 no.1]. The condition that Δ is closed is the condition used in algebraic geometry for a separated scheme (see [Hartshorne, Ch. II Sec 4] and Macdonald (1.11) in [Carter-Segal-Macdonald, LMS Lecture Notes]).

By Theorem 1.1e, Hausdorff spaces are spaces such that limits are unique, when they exist.

For the filters and topology pages: Define base of a topology and base of a filter? Or just go by topology generated by and filter generated by? Then put other stuff in exercises.

The treatment of metric spaces and completion follows [BR] Chapter 2 Exercise ??.

References

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR1726779.

[BR] W. Rudin, Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill 1976. MR0385023.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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