Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 18 March 2012


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 and X. A continuous function f:XY is a function such that if VY is open in Y then f-1(V) is open in X. The morphisms in the category of topological spaces are the continuous functions.

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 0 such that

  1. If x,yX then d(x,y) =0 if and only if x=y,
  2. If x,yX then d(x,y) = d(y,x).
  3. If x,y,zX then d(x,z) 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, 𝒪X) where X is a topological space and 𝒪X is a sheaf of rings on X. The sheaf 𝒪X is the structure sheaf of the ringed space (X, 𝒪X).

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 Cr-manifolds.


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

[Top] A. Ram, Topology at

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

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