Lecture 1 (Sep 8)
We discussed the big idea of the course, and motivated it with a couple of puzzles and a visual proof of the Fundamental Theorem of Algebra.
Lecture 2 (Sep 13)
We introduced the notion of a metric space, giving nine examples of intuitive (and unintuitive) spaces. We finished by discussing--on an informal level--the notion of open and closed, and how one might formulate them in a general metric space. We will take this up rigorously next class.
Lecture 3 (Sep 15)
We came up with formal definitions for open and closed sets in a general metric space. This forced us to define the notion of a boundary of a set, which in turn forced us to define the notion of an (open) ball. Along the way we considered examples of balls in different spaces and open and closed sets in different spaces. We saw that even in situations where the space and metric were familiar, the notion of open or closed might be unintuitive.
Lecture 4 (Sep 20)
We continued discussing metric spaces, in particular convergence of sequences with a metric space. This led us to formulate the notion of the closure of a set, which, intuitively, consists of all the points that are close to the set.
Lecture 5 (Sep 22)
Today we began topology proper. Given a space that doesn't have a natural metric on it---can we develop analysis on it? At first glance, the answer is clearly no, since the main objects of study in analysis (e.g. convergence of sequences, or continuity of functions, or notions like open / closed) are all defined in terms of distance. Our main insight was that actually, all these are defined in terms of something much more basic: a notion of closeness between a given point and a give set. This led us to try to define what closure might mean in the absence of a metric; we came up with properties we might hope any closure operator possesses, and then defined closure to be any operator satisfying all our properties. We saw that this was equivalent to defining a notion of closed, which itself is equivalent to defining a notion of open. Thus, to do analysis on a space, it suffices to produce a list of all the open sets in the space; this list is called a topology.
Lecture 6 (Sep 27)
Motivated by our work last class, we defined a topological space. We constructed a number of examples and non-examples of topological spaces, like the discrete topology, the indiscrete topology, the ray topology on ℝ, the cofinite and cocountable topologies, and the partcular point topology. Some of these topologies were comparable, some were not, and we discussed the notion of refinements (or coarsening) of a topology. Next, inspired by the structure of open sets in ℝusual, we formulated the concept of a basis of a topology. We came up with two necessary conditions for the basis to generate a topology; it turns out these are also sufficient! Finally, we presented Furstenberg's proof of the infinitude of primes using an interesting topology on ℤ.
Lecture 7 (Sep 29)
As practice with the idea of a basis, we introduced the lower limit topology; ℝ with respect to this topology is called the Sorgenfrey line. We were then led to a discussion of a different way to generate a topology, which led us to formulate the notion of a subbasis. Finally, we turned to sequences and convergence in a topological space. After playing around with the definition in metric spaces, Saad conjectured a definition for convergence of a sequence that would make sense in a topological space; we then proved that in a metric space, this definition was equivalent to the usual one. However, it led to some counterintuitive sequences converging. We explored sequence convergence in the Sorgenfrey line, in ℝ with respect to the discrete topology, and finally in ℝ with respect to the particular point topology.
Lecture 8 (Oct 4)
We continued our discussion of convergence of sequences. Trying to understand why convergence is so messed up led us to formulate some nice conditions we could impose on a topological space to make convergence nicer; in particular, we invented the definitions of T0, T1, and T2 spaces (the last of which is usually called Hausdorff). In addition to numerous examples, we proved that in a Hausdorff space, sequences converge to at most one point.
Lecture 9 (Oct 6)
Although T1 spaces aren't ideal for convergence---a sequence might converge to multiple points---it's still relatively nice in that being T1 is equivalent to singletons being closed. Most topological spaces one encounters in the wild are Hausdorff, but there's at least one very famous exception: the Zariski topology. I explained what the Zariski topology on ℝn looks like, and why it's a natural topology to impose (despite not being Hausdorff). We then continued our development of analysis in abstract topological spaces by inventing definitions of interior, boundary, and continuity.
Lecture 10 (Oct 13)
We continued discussing continuity, in particular examining a number of examples. We saw that some functions that look continuous are not, while some functions that do not look continuous are. This led us to formulate a general philosophy that continuity measures not only how nice a function is, but how refined or coarse the domain and codomain are. We then turned to another important topic: connectedness. After considering some examples, we invented a definition that works well (once we got past some of the counterintuitiveness of the subspace topology). We then formulated some equivalent conditions for a space being connected, and considered a bunch of examples.
