MATH 431, Section 1: Topology

As ever, here's a bad joke to start things off:

The function party is going smoothly: the music is groovy, the food is great, and everyone is having a good time and getting along... everyone except for e^x, who is just kind of hanging out in the corner gloomily. tan(x), whose party it is, sees him sulking and decides to try to cheer him up.

"Hey, everybody!" she shouts, "how about we all differentiate?" Cheers rise from all around, and everyone gets in on the action, differentiating themselves, taking first derivatives, second derivatives, using the Chain Rule...there is even some implicit differentiation going on on one side of the room.

But e^x is still just sitting there morosely. tan(x) is disappointed, so she goes over to him. "Hey," she says consolingly, "why not join in on the fun? See, everyone is differentiating, you could do it too!"

"Why bother?" asks e^x, "it won't change anything."

Want class resources? Here ya go:

Syllabus and course description
Homework assignments, by date
Here is a stylesheet that will give you an idea as to what I look for in high-quality mathematical writing.

Resources for using the LaTeX typesetting language:

Getting started with LaTeX: where to find the software you need online
The basics of mathematical typesetting: a LaTeX-created PDF document with many useful math symbols
The basics of mathematical typesetting (raw code): the .tex file corresponding to the previous .pdf file
Using LaTeX to typeset math papers: a LaTeX-created sample math article
Using LaTeX to typeset math papers (raw code): the .tex file corresponding to the previous .pdf file
Including graphics files in LaTeX
Including graphics files in LaTeX (raw code)
Links to free LaTeX compilers:
- MiKTeX (for PCs)
- MacTeX (for Macs)
Here is a link to a free LaTeX text editor: TeXMaker; it works for all computer platforms, but Mac users will probably just want to use the TexShop front end that comes with MacTeX when downloaded from the link above.
Coming soon: a LaTeX template for your "textbook" submissions

Day-by-day goings-on:

Tuesday, January 12th. First day of class! We will spend the first fifteen minutes or so hitting the high points of the syllabus (see above), and then dive right into the first handout, on topological spaces and continuous functions.
Thursday, January 14th. No class: I will be out of town at the Joint Meetings (along with about a third of you!).
Tuesday, January 19th. We may spend a little more time on the first handout, and will almost certainly start the second one, on bases and comparison of topologies.
Thursday, January 21st. After our first round of homework presentations, we will continue with the second handout and being the third, considering examples of topological spaces.
Tuesday, January 26th. We will continue with the handout full of examples.
Thursday, January 28th. After homework presentations, we will finish off the third handout and being the fourth, addressing the product topology.
Tuesday, February 2nd. We spent about half of the the period discussing Homework 2, and the other half revisiting product topologies (see the previous entry for the notes).
Thursday, February 4th. We will wrap up our discussion of the product topology and begin talking about subspace topologies.
Tuesday, February 9th. More on subspace topologies! We will also spend a little bit of time reviewing basic definitions with this handout.
Thursday, February 11th. We continue with our next handout, on closed sets.
Tuesday, February 16th. We should be able to finish up the first handout on closed sets; see the link above! We may even get started on the next handout, concerning limit points.
Thursday, February 18th. We began talking through the handout on limit points; see the entry above for the handout.
Tuesday, February 23rd. We will finish up our handout on limit points, and begin talking about what in the heck continuity has got to do with convergence, with this handout.
Thursday, February 25th. We will wrap up the handout above on convergence and continuous functions. We are just about ready to talk about connectedness, compactness, and completeness. But first, a detour into into a field that will feel familiar to many of you, metric spaces.
Tuesday, March 2nd. SNOW DAY!
Thursday, March 4th. More on metric spaces.
Tuesday, March 16th. We will finish up our first handout on metric spaces and move on to the next, in which you will encounter a nontrivial example of a metric space.
Thursday, March 18th. We continue on the second metric space handout. Next week we begin talking about compactness and completeness!