Commentary by Chris Pressey

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Topology

Introduction to topology

• rating: 1

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Counterexamples in Topology

• rating: 3

Don't worry if you don't know topology — it's not the topology that makes this a worthwhile read, it's the counterexamples.

The Lattice of Topologies: Structure and Complementation

• rating: classic

The set of all topologies on a set forms a lattice where the discrete topology is the supremum and the trivial topology is the infimum.

I figured this must be the case before I found this paper. I also figured it must be an established result. I enjoyed locating the paper in which it was established.

Finite Topological Spaces

• rating: 2

Lecture notes. All told, similar to [A Short Study of Alexandroff Spaces][], but goes into more detail. Good quote:

There is a hierarchy of “separation properties” on spaces, and intuition about finite spaces is impeded by too much habituation to the stronger of them.

This seems to lead people to believe that finite topological spaces are uninteresting (because they expect that the interesting things about any topological space revolve around the seperation properties, apparently).

That might not be quite it. Finite topologies probably just don't "look very much like topologies" to people who study the more usual sort of topologies. They look more like... lattices?... or something out of order theory, anyway. If you have some open sets, union and intersection will both get you more open sets.

Made more concrete by Lemma 1.15 "A finite set X with a reflexive and transitive relation ≤ determines a topology", and this quote from Section 6, "We have seen that enumeration of finite sets with reflexive and transitive relations ≤ amounts to enumeration of the topologies on finite sets."

Section 3 does some combinatorics, looking at all the possible topologies on a 3-point set and a 4-point set.

Section 4, on connectedness and path-connectedness, tells us that finite spaces are "surprisingly richly related to the “real” spaces that algebraic topologists care about." There are maps from topologies on subsets of the reals, to finite topologies.

Sections 5 and 6 talk about homotopies and homotopy equivalences.

A Short Study of Alexandroff Spaces

• rating: 1

A topology is Alexandroff when arbitrary intersections of open sets are open.

Interesting finite topologies are usually Alexandroff. T_1 and above (e.g. Hausdorff) topologies are uninteresting, if they're finite. The section on "Hausdorff Alexandroff spaces" demonstrates this by giving a proof that a space is Hausdorff Alexandroff iff it is discrete.

Alexandroff spaces are not necessarily finite, but they are "finitely based".

Products and quotients of Alexandroff spaces are Alexandroff spaces.

I don't know how interesting Alexandroff spaces are.