Hausdorff and separable spaces

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

Last updates: 20 May 2011

Hausdorff spaces

Let $X$ be a set. The diagonal map is

$\Delta :X\to X\times X\text{given by}\Delta \left(x\right)=(x,x)$.

The image of the diagonal map $\Delta \left(X\right)$ is the graph of the identify function ${\mathrm{id}}_X:X\to X$.
A Hausdorff space is a topolgical space such that $\Delta \left(X\right)$ is closed in $X\times X$, where $X\times X$ has the product topology.

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

(a)   $X$ is Hausdorff.

(b)   If $x,y\in X$ and $x\ne y$ there there exists a neighborhood $N_x$ of $x$ and a neighborhood $N_y$ of $y$ such that $N_x\cap N_y=\varnothing$.

(c)   The intersection of the closed neighbourhoods of any point of $X$ consist of that point alone.

(d)   for every set $I$, the diagonal of the product space $Y=X^I$ is closed in $Y$.

(e)   no filter on $X$ has more than one limit point.

(f)   If a filter $ℱ$ on $X$ converges to $x$ then $x$ is the only cluster point of $ℱ$.

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 ${\ell }^p$ is separable if $p\ne \infty$ and ${\ell }^{\infty }$ 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 ${ℝ}^k$ is separable.

Notes and References

These notes follow Bourbaki [Bou???] Chapter II???. The condition that $\Delta$ 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]).

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. MR??????.

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

page history