A Basic 5 Minute Crash Course into Basic Topology

Havn't had the pleasure yet. I am sure I have seen glimpses of it but as a formal study, no.

Actually, now that I think or it, I did take a class starring Algebraic Geometry one time. It was ok but not really my thing. At least this class which was very focused on a topic called Gröbner Bases. Gröbner Bases are cool because they allow to you very easily and computationally find solutions to very general polynomials (IE in many dimensions). This has applications to parameterized surfaces. 

Lots of ring theory involved. I have since used Gröbner Bases in my research a few times but don't plan to make a huge career focus out of them.

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Ah ha!! So you were a spy! You say engineer but you really belonged to the Mathematicians camp... Oh well, personanally I am sure I will end up spending my life as a computer programmer.

Where did you study?

Undergrad: Slippery Rock University. LOL.  It was close to home & I was poor.  Grad: Miami of Ohio.  I went there because a math prof at Slippery Rock had a friend on the Miami faculty, so I got an "in" for a TA; didn't even have to take the GRE back then, was still  the "good old boy" system.  I got to go to school for free, plus $100 a week (which I could just barely get by on 78-80) in exchange for teaching 1 class a semester and doing a 3 hour calculus help session 1 night a week.  I taught a 5 credit class, Algebra and Trig, which I suppose now-a-days would be called pre-calculus. Oh, youth!  Miami didn't have a Ph.D program then, so I decided to get a job.  Mistake I think; should have went somewhere else and got that post hole digger degree.  :P  So I ended up being an engineer all this time; not nearly as engaging as theoretical mathematics, especially once I got into management; what a bunch of hokum.  Regrets, I've had a few.....lolz

I don't think Albegraic geometry would count, I am pretty sure they are a bit different but hey, I'm not discriminating!

The term non-commutuative in Non-Commutative geometry suggested to me that it falls under the umbrella of Algebraic Geometry. Again though, I really don't know a lot about that field.

Yes, that is, in my view, the proper definition of a topology (that subset of the power set)

It's been a long long time, but I really liked topology, so I at least remember some definitions. :P  I enjoyed the progression of vague concepts of separation without any sort of metric what-so-ever, from Hausdorf to Normal; I don't know why, I just found it aesthetically pleasing for no particular reason, just in and of itself.  Where is my old textbook?  lol.  I liked groups and rings and fields too, but topology was such a refreshing change from Abstract Algebra, at least for me, and I can't even say why.  I bet there is a free textbook on the web somewhere.  600 level real analysis was finally some stuff that was difficult for me.  And Complex Analysis, integrating around contours, I didn't like that at all. :P  So what did they grill me about in my 3 hour final oral exam?  The topology of sets of analytic functions in the complex plane.  That was rough for me. :P  I can't even define most of those terms now. Use it or lose it I guess.

Here's a good one on Algebraic Topology:

http://www.math.cornell.edu/~hatcher/AT/ATpage.html

Complex analysis was really fun for me because when I took it, it was taught by one of the most hard core topologists at my University (The rest of the class was not quite as enthusiatic though).

A desent general topology book which works to build intuition using metric spaces first and topo spaces second is:

http://www.amazon.com/Introduction-Topology-Theodore-W-Gamelin/dp/0486406806/ref=sr_1_2?ie=UTF8&s=books&qid=1265060526&sr=8-2

Its also a dover book, so its affordably priced.

Thanks!!!  Probably best to start with metric spaces, have that analog of distance to hang on to at first.

I had a great guy for complex analysis, but there was a language barrier that really hurt.  I think I got lazy and didn't put the work in myself that I should have.  They picked up on that right away in that final all encompassing oral exam too, which is no doubt why they grilled me on it. They made me prove all kinds of odd stuff they made up on the spot; I have no idea what any more, weird topologies defined on sets of specific analytic functions and then some, but they really zeroed in on my weaknesses, right off the bat.   3 hours of sweat, blood and chalk as I recall.  But I did it.  Phew.  Sure couldn't now.

I apologize for veering off topic with personal recollections.  It's just that these topics bring back a lot of really old memories I haven't thought of in a long time, the good old days and all that.

Reminisce all you want. No judgement here! I like hearing about it. Ever consider going back for that Piled Higher and Deeper(Ph. D.)?

I saw my first phd comics 6 months ago or so...relatively new to them still. I like them a lot though.

HA YES!!! I love Ph. D. Comics. Look for the one about his visit to CERN- it is so true!!

Considered, yeah. hmmmm.  Man I would really have to bone up on all the stuff I forgot, which is most stuff. :P  APPLIED maybe?  I am so tired of statistics though.