Gracious Living

To Do

This page will list all the topics I’m planning to cover.  If the blog gets large enough that navigating it becomes confusing, I might also put up a list of the topics I have covered in a nice, organized fashion.  If there’s something you are curious about or want to see explained, just comment here or on some post and I’ll add it to the docket!

Short term:

  • Set Theory
    • Ordinal/cardinal arithmetic?
    • Construction of \mathbb{Z},\mathbb{Q},\mathbb{R}
  • Topology
    • Kuratowski closure operators
    • Quotient topologies
    • Compactness
    • Countability and separation axioms
    • Complete metric spaces
  • Algebra
    • Groups: solvability
    • “standard”/combinatorial finite group theory — Sylow theorems, conjugacy classes, etc.
    • Rings
    • Fields

Long term:

  • Point-set topology: continuity, connectedness, compactness, separation, homotopy
  • Group theory
  • Algebraic topology: homotopy, homology, cohomology
  • Banach-Tarski paradox
  • Higher homotopy groups are abelian

3 Comments so far
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Hi, I’m particularly interested in Banach-Tarski paradox, which contradicts intuition very much. However, I’m a engineering guy, so I don’t have luxuries of things like group theory, etc. to understand this topic, though I do try to teach myself some real analysis. So, may I suggest you open a series of posts, starting from basic knowledge of topology, groups, and finally leading to the explanation of Banach-Tarski paradox ?

Comment by Qiang

Sure thing! I actually think we can get to it pretty soon, which I’d like to do since it’s a very cool paradox. Since it’s about \mathbb{R}^3, we don’t need that much topology beyond what we already know. The group theory is going to be slightly more important. I introduced groups yesterday and I’m going to keep posting about them for a while. If you’re unfamiliar with countability, I suggest you read the post I wrote about it (and possibly the preceding one about cardinality), which will be important to the proof. Thanks for reading!

Comment by thetwomeatmeal

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