Topology

Jan 4, 2017

I have a friend who is generally a fan of mathematical structures and the relationships between them. For Christmas, he asked me to make a diagram of certain sets of mathematical objects and the relationships between them. One instance of this would have been a diagram of complexity classes, with an arrow from class C to class D if every problem in class C was also in class D. Another instance would be a diagram of types of algebraic objects (monoids, groups, rings, etc.), with arrows indicating facts such that all groups are monoids. Instead, I chose to diagram types of topological spaces - plotting properties involving separation, compactness, connectivity, and metrisability, as well as which properties implied which other properties. I also wrote definitions that could theoretically be understood by anyone who understood set theory and equivalence relations, some theorems that hopefully provoke interest in these properties, and some example topological spaces to classify.

Here is the diagram (dot file), the definitions (tex), the theorems (tex), and the example spaces (tex). In the process of researching topological facts to include, I also found a wiki specifically about topology, and a website that is a search engine for topological spaces.

I hope you enjoy them, and if you spot any errors, please email me and let me know.