Hausdorff Spaces

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

and

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

Last updates: 15 September 2009

Hausdorff spaces

A Hausdorff space is a topological space $X$ such that if $x, y \in X$ and $x \neq y$ then there exist a neighbourhood $N_{x}$ of $x$ and a neighbourhood $N_{y}$ of $y$ such that $N_{x} \cap N_{y} = \emptyset$ .

Let

X

be a topological space. Show that [THM OR EXERCISE?] the following are equivalent:

Any two distinct points of $X$ have disjoint neighbourhoods.
The intersection of the closed neighbourhoods of any point of $X$ consist of that point alone.
The diagonal of the product space $X \times X$ is a closed set.
For every set $I$ , the diagonal of the product space $Y = X^{I}$ is closed in $Y$ .
No filter [NOT YET DEFINED] on $X$ has more than one limit point [NOT YET DEFINED].
If a filter $ℱ$ on $X$ converges to $x$ , then $x$ is the only cluster point [NOT YET DEFINED] of $x$ [SHOULD THIS BE X OR x]

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)