Spaces

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

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

Last updates: 11 November 2009

Spaces

A topological space is a set $X$ with a collection of open subsets of $X$ which is closed under unions, finite intersections and contains $\varnothing$ and $X$.

Let $X$ and $Y$ be topological spaces[??? - I added this sentence]. A continuous function $f:X\to Y$ is a map such that $f^{-1}\left(V\right)$ is open in $X$ for all open subsets $V\subseteq Y$. The morphisms in the category of topological spaces are continuous functions.

Let $X$ be a topological space. Recall the following definitions.

1. A closed subset of $X$ is a set which is the complement of an open set in $X$.
2. The space $X$ is compact if every open cover has a finite subcover.
3. The space $X$ is locally compact if every point has a neighbourhood with compact closutre.
4. The space is totally disconnected if there is no connected subset with more than one element.
5. The space $X$ is Hausdorff if ${\Delta }_X=\left\{\left(x,x\right)|x\in X\right\}$ is a closed subspace of $X\times X$, where $X\times X$ has the product topology.

These definitions rely on earlier definitions which can be found here [??? - link to notes page].

The topological space $X$ is Hausdorff if and only if for any two points in $X$ there exist neighbourhoods of each of them which don't intersect.

A metric space is a set $X$ with a metric $d:X\times X\to {ℝ}_{\ge 0}$ such that ... [??? - missing end of sentence?] A Cauchy sequence is a sequence $\left(p_i\in V|i\in {\mathbb{Z}}_{>0}\right)$ such that, for every positive real number $\epsilon$ there is a positive integer $N$ such that $d\left(p_n,p_m\right)<\epsilon$ for all $n,m,>,N$. A sequence $\left(p_i\in V|i\in {\mathbb{Z}}_{>0}\right)$ converges if there is a $p\in V$ such that, for every $\epsilon \in {ℝ}_{>0}$, there is an $N\in {\mathbb{Z}}_{>0}$ such that $d\left(p_n,p\right)<\epsilon$ for all $n>N$. A metric space is complete if all Cauchy sequences converge.

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)