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 $\emptyset$ and $X$ . A continuous function $f : X \to Y$ is a function such that if $V \subseteq Y$ is open in $Y$ then $f^{- 1} (V)$ 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 $X \to pt$ 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 $Δ_{X} = {(x, x) | x \in X}$ 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 $d : X \times X \to ℝ_{\geq 0}$ 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 $d (x, z) \leq d (x, y) + d (y, z)$ .

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

Manifolds, Varieties and Schemes

A ringed space is a pair $(X, 𝒪_{X})$ 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 $(X, 𝒪_{X})$ .

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 $ℝ^{n}$ .
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 $C^{an}$ 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.

[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) 691-732 MR2045836 arXiv:math/0209133v3

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