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Filters in Topology

Blogs > Plexa
Post a Reply
1 2 Next All
Plexa
Profile Blog Joined October 2005
Aotearoa39261 Posts
December 16 2010 19:13 GMT
#1
For those of you who don't know, I'm currently studying for my Honours in Mathematics. Part of that comprises a dissertation we have to write on a research topic (not necessarily new research). For my project I am working in an area of Topology which deals with Convergence Spaces (it's like a generalised topological space).

I am currently in the process of writing up the work I have done over the last semester (it is a two semester project, one semester remaining to finish it!). After about a week of slugging away at things, I have finally produced the first draft of the first section of my dissertation - and well, it feels damn good !

As with everything in my life, I thought I would share it with TL for those few people that might be remotely interested.

This section of my dissertation concerns Filters. They are a kind of generalised sequence in arbitrary topological spaces. They are the fundamental object of a convergence space and hence studying them in a topological setting is necessary before proceeding into the more general setting of a convergence space. There are three parts to this section: the first part concerns general definitions and some key results, the second concerns an application to the Riemann Integral and the last concerns an application to compactness (with the slickest proof of tychonoff's theorem on the product of compact topological spaces you can find!)

Some general notes:
- The introductory paragraphs to each section will be getting rewrites
- There is one proof in the integration section which appeals to a theorem not proven in the paper, it is noted and I should probably include the theorem
- If you spot any errors please let me know latex can be a bitch some times!

Link: http://www.teamliquid.net/staff/Plexa/Dissertation/filters.pdf

Enjoy!

Administrator~ Spirit will set you free ~
Bibbit
Profile Blog Joined December 2009
Canada5377 Posts
December 16 2010 19:19 GMT
#2
I would read it because I've always got the impression that you're a really smart guy but I wouldnt understand anything. Math has always frightened me.

But best of luck with your dissertation though. :D:D:D
wherebugsgo
Profile Blog Joined February 2010
Japan10647 Posts
December 16 2010 19:53 GMT
#3
Haha, I'm taking the honors introduction to analysis course here at Berkeley, and we covered filters a month or so ago. Tomorrow is our final >_<

http://math.berkeley.edu/~wodzicki/H104.F10/Topology.pdf

Filters start on page 4, but you already know all of this stuff.

Good luck with your dissertation!
Plexa
Profile Blog Joined October 2005
Aotearoa39261 Posts
December 16 2010 19:58 GMT
#4
Ah gl for your final! You must have covered nets as well I fortunately got to skip all of that haha
Administrator~ Spirit will set you free ~
vAltyR
Profile Blog Joined July 2008
United States581 Posts
December 16 2010 19:58 GMT
#5
Very interesting. I wish I understood it. I don't know very much about topology, so most of this feels way over my head, even as a math major myself. The LaTeX looks fine though, from what I can tell.
내 호버크라프트는 장어로 가득 차 있어요
vvSiegvv
Profile Joined March 2010
United States364 Posts
December 16 2010 20:32 GMT
#6
Wish I could understand this, but I haven't gotten into topology at all yet, but it sounds pretty interesting. (2nd year math/compsci major)

Best of luck Plexa!
EsbenPM
Profile Joined April 2006
Denmark364 Posts
December 16 2010 20:35 GMT
#7
Looks interesting although i don't really understand much of it. But good luck anyway on your future work.

Oh, and making that look good in Latex must have been a pain in the ass with all those special characters and math expressions
Hi
LightRailCoyote
Profile Blog Joined January 2008
United States982 Posts
December 16 2010 20:36 GMT
#8
going to read it now.
AKA SurfSolar ----- This is the product of a DIY inadequate home
micronesia
Profile Blog Joined July 2006
United States24690 Posts
December 16 2010 20:47 GMT
#9
I'm not exactly sure what an Honours in Mathematics is equivalent to in the systems I'm familiar with. After you finish your dissertation, what will that make you?

I'd give you feedback but I'm too unfamiliar with this type of math to be much help.
ModeratorThere are animal crackers for people and there are people crackers for animals.
PrinceXizor
Profile Blog Joined April 2010
United States17713 Posts
Last Edited: 2010-12-16 22:07:21
December 16 2010 22:05 GMT
#10
Now i may be an english major, but i've got to tell you there apparently is an error in the first sentence:
It quickly becomes apparent that sequences are not adequet in a general topo-
logical setting and that new more general concepts need to be introduced.

Adaquate is spelled wrong, and the latter half: "and that new more general concepts need to be introduced." should either have the e in new changed to now, or have a comma after new.

"and that new, more general concepts need to be introduced".

But all of the math gibberish seems very intelligent. :D i noticed a few more errors that may or may not be errors depending on gibberish context. are you in the proofreading stage yet?

EDIT: and also this is a rather dull and confusing beginning for the paper anyway. perhaps explain what makes it apparent, something like "Upon using some gibberish with another thing, it becomes apparent..." ect.
Pokebunny
Profile Blog Joined June 2008
United States10654 Posts
December 16 2010 22:55 GMT
#11
I thought it was adequate, not adaquate.
Semipro Terran player | Pokebunny#1710 | twitter.com/Pokebunny | twitch.tv/Pokebunny | facebook.com/PokebunnySC
PrinceXizor
Profile Blog Joined April 2010
United States17713 Posts
December 16 2010 23:02 GMT
#12
On December 17 2010 07:55 Pokebunny wrote:
I thought it was adequate, not adaquate.

it is. epic fail by me. my bad.
Laconic
Profile Joined June 2010
United States32 Posts
Last Edited: 2010-12-16 23:07:27
December 16 2010 23:07 GMT
#13
Muphry's Law is inerrant.

And wow, nice job, Plexa. I don't even begin to understand what you are writing about, but I am impressed! It is like some sort of mysterious, majestic poetry written by Martians.
blahman3344
Profile Blog Joined March 2009
United States2015 Posts
Last Edited: 2010-12-16 23:29:39
December 16 2010 23:28 GMT
#14
I was able to understand it (with my inferior math knowledge) until the basic concepts part, so I'd say that you did a good job writing your draft. =)
Though I have to say, if that first part is really that long, I can only imagine how long the rest of the dissertation is going to be. ^^

well good luck Plexa, I'm sure you'll do fine. =P
I like haikus and / I can not lie. You other / brothers can't deny
]343[
Profile Blog Joined May 2008
United States10328 Posts
December 17 2010 00:02 GMT
#15
Hmm, about math writing:

Please end your sentences with periods! Even statements of "remarks"/"lemmas" etc.. (and equations are sentences (or parts thereof) too, so if they end a sentence, use a period!" (You end sentences with periods like... half the time.) [E.g. Lemma 1.44 should have a period after the (IV). -L(f,P) = U(-f,P). And perhaps commas after the other equations.]

You could also try to use the \begin{eqnarray} environment for that, though I suppose that's been deprecated... rather, use the \begin{align} environment. This gives you equation numbers on each line (after the equation, though... so if you want the number to go before the equation, just stick with this. And maybe use \textrm to make it non-math-mode.) Automatically numbered equations
can be annoying to keep up with, but you can do like \label and \ref and funny things like that to keep track of it dynamically (it requires compiling twice though.)

Try not to form run-on sentences. Use commas or semicolons (or even periods) as necessary. (As a general rule, if the sentence has more than one clause, it should have a comma or semicolon!) Also, fix typos. For example, Remark 1.49: "havenn't." Also in Remark 1.49, "implicitly" would be better than "silently," I feel.

Too bad I don't understand all the math itself though ... maybe I'll take topology in a year or two though, so yay
Writer
Plexa
Profile Blog Joined October 2005
Aotearoa39261 Posts
December 17 2010 02:04 GMT
#16
Thanks very much for all of that ]343[, I'm not going to touch this thing today since it would be terrible for my sanity, but when I get to reviewing it tomorrow or next week I'll incorporate the things you suggest.
On December 17 2010 05:47 micronesia wrote:
I'm not exactly sure what an Honours in Mathematics is equivalent to in the systems I'm familiar with. After you finish your dissertation, what will that make you?
At the moment I have a Bsc in Maths, after I'm done with this year I'll have a Bsc (hons) in Maths. The honours year comprises of 6 papers and 1 year long dissertation. It's not expected to be groundbreaking or revolutionary in anyway, it's purely for the research experience. After this I will most likely begin my Masters (that would be in july next year) and then go from there.
Administrator~ Spirit will set you free ~
micronesia
Profile Blog Joined July 2006
United States24690 Posts
December 17 2010 02:10 GMT
#17
Okay that makes a bit more sense. I was looking through and this document is pretty dense but not substantive enough to be the type of dissertation that would get you a doctorate :p
ModeratorThere are animal crackers for people and there are people crackers for animals.
Plexa
Profile Blog Joined October 2005
Aotearoa39261 Posts
December 17 2010 02:53 GMT
#18
Yeah this is the baby stuff anyway, all background information haha
Administrator~ Spirit will set you free ~
See.Blue
Profile Blog Joined October 2008
United States2673 Posts
December 17 2010 03:17 GMT
#19
Putting off studying for my analysis final as well and saw this haha. Latex makes everything look so pretty. Great job so far!
Klockan3
Profile Blog Joined July 2007
Sweden2866 Posts
Last Edited: 2010-12-17 04:00:14
December 17 2010 03:59 GMT
#20
F.3 is wrong, it should be "then" and not "and".
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