Spaces
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 18 March 2012
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 ${\Delta}_{X}=\left\{\right(x,x\left)\phantom{\rule{0.5em}{0ex}}\right\phantom{\rule{0.5em}{0ex}}x\in X\}$ is a closed subset of $X\times X$,
where $X\times X$ 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
$d:X\times X\to {\mathbb{R}}_{\ge 0}$
such that

If $x,y\in X$ then
$d(x,y)=0$
if and only if $x=y$,

If $x,y\in X$ then
$d(x,y)=d(y,x)$.
 If $x,y,z\in X$
then $d(x,z)\le d(x,y)+d(y,z)$.
A metric space is complete if all Cauchy sequences converge in
$X$.
Manifolds, Varieties and Schemes
A ringed space is a pair $(X,{\mathcal{O}}_{X})$
where $X$ is a topological space and
${\mathcal{O}}_{X}$ is a sheaf of rings on $X$.
The sheaf ${\mathcal{O}}_{X}$ is the structure sheaf
of the ringed space $(X,{\mathcal{O}}_{X})$.

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
${\mathbb{R}}^{n}$.

A smooth manifold is a ringed space that is locally isomorphic to
${\mathbb{R}}^{n}$
(using the structure sheaf
${C}^{\infty}$ on
${\mathbb{R}}^{n}$).

A topological manifold is a ringed space that is locally isomorphic to
${\mathbb{R}}^{n}$
(using the structure sheaf
$C$ on
${\mathbb{R}}^{n}$).

A ${C}^{r}$manifold is a ringed space
that this locally isomorphic to ${\mathbb{R}}^{n}$
(using the structure sheaf
${C}^{r}$ on
${\mathbb{R}}^{n}$).

A complex manifold is a ringed space that is locally isomorphic to
${\u2102}^{n}$
(using the structure sheaf
${C}^{\mathrm{an}}$ on
${\u2102}^{n}$).
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. 12], [AM, Ch. 1 Ex. 1920 and Ch. 6 Ex. 512] 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.
[Top]
A. Ram, Topology at
http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html.
[Le]
Dual canonical bases, quantum shuffles and $q$characters,
Math. Zeitschrift 246 (2004) 691732
MR2045836
arXiv:math/0209133v3
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