Paracompact Spaces

Toroid, via Wikivisual


It’s strange that I was actually researching paracompact topological spaces on Sunday and that we are seeing those types of spaces in my complex analysis class. We just started the Berenstein & Gay Complex Variables¹ book and things are pretty interesting. I actually deduced that we were heading there because of some of the concepts that we are seeing.

A paracompact space is a topological space in which every open cover admits an open locally finite refinement.


I spent some time this weekend working on Riemann surfaces, meromorphic functions, as well as étale spaces and sheaf cohomology. I’m getting the basics down. I’m still really interested in category theory, differentiable manifolds, as well as complex and conformal manifolds. Germs, sheaves, presheaves and stalks are really quite interesting. I need to work more at my classes and at my own research. For now, the research is just about learning new concepts. Strangely enough, a lot of what I’m researching ties into my complex analysis class¹.

This week, I want to formalize two research assistanships. They will most likely be in PDEs and Optimization. Although these subjects interest me less, I’m keen to get my name in academic papers. This would be the way to go. There is a chance that one might be in differential geometry, which would really be cool.

I’ll also have to start preparing my candidacy for more prestigious research assistanships, one which is due for January ’10. I want to nail that one. I guess that this is the way that research starts, by being curious about certain topics, investigating them yourself and finding out what has been published about them.

I only spent about 4 hours on research today. I need to make more of an effort during the week. It’s easy to see that when I’m most relaxed, that’s when my curiosity and interest really starts up. Anyway, I need to finish my real analysis exercises. I’ve still got 35 to finish. I’ve also got 15 modern algebra problems to complete. Since Friday, I’m more relaxed, so I have no doubt that I’ll be ready in time. I also need to start reviewing theorems and definitions for the real analysis midterm next week. In modern algebra, we will be doing a take-home exam. I think we’ll have a week to finish it.

My basic tool for now is my academic journal. It’s a A91 Blueline 200p hardcover notebook²
that does the job pretty well. It’s lined, compared to the Moleskines I use for class notes, and the notes follow the same pattern I’ve established with my other ones. I rely on a color-coded system for definitions, theorems, and properties.While I’m researching topics, I get very focused. So it’s a good idea to review what has been done and to take some extra notes on what to study next.


I need to do some needed maintenance on my computer. Some of the HDs that broke down over the last few years must still be on warranty. I should send them back. Most of my HDs are from Western Digital, because of their reliability and their exemplar customer service. Warranties on HDs last 3 years.

The other thing that I need to do is to format my main drive so that I can install Win 7. I’ve been wanting to do that for a few months now. The thing is that I need to buy another HD to migrate the data to. My main data drive broke down in June, about a week before I came back to Taiwan.

As I mentioned before, I’ll get the equivalent of a 2011 iMac as well as a MBP before I leave. I plan on toughing things out with my desktop and ASUS WFM laptop for now. One thing is for sure, I want to boost the laptop to max specs. It also needs to be serviced. It runs very hot and I noticed that it shuts down after about 10 minutes on the highest screen brightness. I have to keep it on a lower setting, which is annoying.

I’m hoping by that time that there will be a 30″ or 35″ iMac. This is a perfect all-in-one solution. I don’t do games so I don’t need a PC all that much. The price of the iMac is about the same as a big monitor. The Apple Cinema display, which is 30″, costs about $1,800. For almost the same price, you get a 27″ screen, with an integrated computer. It makes a lot of sense and it’s somewhat portable, so I’ll be able to bring it back to North America. Sure, 1TB isn’t enough storage, but I’ll probably get myself an external USB drive array, something around 4-8TB should to the job. Something like a Drobo filled with 2TB WD drives should do the job.  The good thing about Drobo is that you can start with say 2×2TB, which wouldn’t cost that much. Probably $300 in HDs plus another $300 for the Drobo. It would probably be smart and start this ASAP, as I need more HDs, but don’t want to install them into my computer.

* * * * *

[¹]: Berenstein & Gay, Complex Variables, 1991, Springer Verlag, GTM Vol 125.
[²]: I’ve decided to use this instead of getting another notebook. Only three pages were used, and this dates back to January ’08.

Author: range

I'm mathematician/IT strategist/blogger from Canada living in Taipei.

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