## Spaces

A topological space is a set $X$ with a specified collection of open subsets of $X$ which is closed under unions, finite intersections and contains $\varnothing$ and $X$. A continuous function $f:X\to Y$ is a function such that if $V\subseteq Y$ is open in $Y$ then ${f}^{-1}\left(V\right)$ is open in $X$. The morphisms in the category of topological spaces are the continuous functions.

• A closed subset of $X$ is the complement of an open set of $X$.
• The space $X$ is quasicompact if the function $X\to \mathrm{pt}$ is proper.
• The space $X$ is compact if it is quasicompact and Hausdorff.
• The space $X$ is locally compact if every point has a neighbourhood with compact closure.
• The space $X$ is connected if there does not exist a
• The space $X$ is totally disconnected if there is no connected subset with more than one element.
• The space $X$ is Hausdorff if ${\Delta }_{X}=\left\{\left(x,x\right)\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}x\in X\right\}$ is a closed subset of $X×X$, where $X×X$ has the product topology.
• The space $X$ is irreducible if $X$ is nonempty and every pair of nonempty open subsets intersect.
• The space $X$ is Noetherian if the closed subsets of $X$ satisfy the descending chain condition.

HW: Show that a topological space $X$ is Hausdorff if and only if for any two points in $X$ there exist neighbourhoods of each of them that do not intersect.

HW: Show that a topological space $X$ is quasicompact if and only if every open cover contains a finite subcover.

A metric space is a set $X$ with a metric $d:X×X\to {ℝ}_{\ge 0}$ such that

1. If $x,y\in X$ then $d\left(x,y\right)=0$ if and only if $x=y$,
2. If $x,y\in X$ then $d\left(x,y\right)=d\left(y,x\right)$.
3. If $x,y,z\in X$ then $d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$.

A metric space is complete if all Cauchy sequences converge in $X$.

## Manifolds, Varieties and Schemes

A ringed space is a pair $\left(X,{𝒪}_{X}\right)$ where $X$ is a topological space and ${𝒪}_{X}$ is a sheaf of rings on $X$. The sheaf ${𝒪}_{X}$ is the structure sheaf of the ringed space $\left(X,{𝒪}_{X}\right)$.

• A scheme is a ringed space that is locally isomorphic to an affine scheme.
• A variety is a ringed space that is locally isomorphic to an affine variety.
• A manifold is a ringed space that is locally isomorphic to ${ℝ}^{n}$.
• A smooth manifold is a ringed space that is locally isomorphic to ${ℝ}^{n}$ (using the structure sheaf ${C}^{\infty }$ on ${ℝ}^{n}$).
• A topological manifold is a ringed space that is locally isomorphic to ${ℝ}^{n}$ (using the structure sheaf $C$ on ${ℝ}^{n}$).
• A ${C}^{r}$-manifold is a ringed space that this locally isomorphic to ${ℝ}^{n}$ (using the structure sheaf ${C}^{r}$ on ${ℝ}^{n}$).
• A complex manifold is a ringed space that is locally isomorphic to ${ℂ}^{n}$ (using the structure sheaf ${C}^{\mathrm{an}}$ on ${ℂ}^{n}$).

## Notes and References

These notes are from ?????. The definitions of Hausdorff, compact and quasicompact spaces follow [Bou, Topology]. See also the web pages ???NOTES??? pages. The definitions of irreducible spaces and Noetherian spaces are found in [Bou, Comm Algebra Ch. II §4 No. 1-2], [AM, Ch. 1 Ex. 19-20 and Ch. 6 Ex. 5-12] and [Mac, Ch. 2] See also the web pages ???NOTES???. [Bou, Variétés] contains a treatment of ${C}^{r}$-manifolds.

## References

[Mac] I.G. Macdonald, Algebraic Geometry: Introduction to Schemes, W.A. Benjamin, New York, 1968.

[Le] Dual canonical bases, quantum shuffles and $q$-characters, Math. Zeitschrift 246 (2004) 691-732 MR2045836 arXiv:math/0209133v3