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Topology Explained

This is a collection of notes on topics in topology. These notes will be useful to people learing topology. This collection is still being developed; please send us your lecture notes or your suggestion for new notes.

These notes were written by Henno Brandsma and Abhijit Dasgupta for the Topology Atlas Q+A Board.

Axiom of choice
A proof of Tychonoff Theorem implies AC
De Morgan in general and choice
Uniform spaces
Some elementary facts on uniform spaces
Dimension
A little overview of dimension
Normality, paracompactness, metrisability
On paracompactness, full normality and the like
Proof of a lemma on paracompactness plus applications
Proof of the Nagata-Smirnov metrisation theorem
Paracompactness, covers and perfect maps
Useful theorems on normal spaces
The shrinking lemma
Limit points of discrete sets in metric spaces
Connectedness
Connectedness I
Components in three product topologies on R^\omega
Boundaries, interiors and open maps
Compactness
Covering maps and perfect maps
Nets, cluster points and the Tychonoff theorem
H-closed and not compact
Inverse limits, compactness and why Hausdorffness is important
Tychonoff and Kolmogorov extension
Compactness in function spaces: Arzela-Ascoli type theorems
Cardinal functions
Arhangel'skii's theorem, a proof
Quotient maps
Quotient maps
General constructions
Embeddings in to products of the Sierpinski space
The fourteen subsets problem: interiors, closures and complements
Countable metric spaces without isolated points
Algebraic topology
Checking all the axioms for the first homotopy group

Revised: June 25, 2005.


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