Topology issues

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There are a variety of unsatisfactory aspects to the general topology notes I 
have. Namely they include:

1. Compactness isn't discussed anywhere at all.
2. Connectedness isn't discussed adequately.
3. The product topology is not given enough explanation.
3.1. The universal property for the product topology is not developed anywhere.
3.2. The tube lemma isn't discussed anywhere.
4. The proof for the pasting lemma is not produced. Nor is a 
motivation/explanation of the pasting lemma.
5. The discussion of continuity is limited. An example of discontinuity should 
be discussed.
6. There are no exercises in this text.

So the solution should involve:
1. Expanding each section considerably, as if they were a chapter.
2. Include exercises and their solutions.
3. Develop a better style to avoid this Bourbakite grocery list writing style.
4. Perhaps start from scratch completely and try to explain the importance of 
topology from a different angle?

Original issue reported on code.google.com by pqnelson on 27 Sep 2010 at 4:56

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Additional topics that should be covered:

1. Metric topology
2. Limit Points
3. Path connectedness, Hausdorff space, and Normal Spaces
4. Countability and Separation Axioms

But a review of set theory should be given in an appendix (or more precisely, 
the appendix should be developed). The Axiom of Choice is rarely given a good 
discussion.

Original comment by pqnelson on 27 Sep 2010 at 5:01

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Original comment by pqnelson on 22 Feb 2011 at 2:09

• Changed state: Started