# 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
• Higher homotopy groups are abelian

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!