Undergraduate Research Topics in Point-Set Topology?

[sutupidmath]

Hi all,

Like the title suggests, i am interested in finding a topic in topology that would serve as the basis for a research paper. Since i am currently taking a first course in Topology (Munkres), i am basically looking for something that is not too advanced. So far i haven't been able to find any article that is aimed for undergraduates in this field... all of them seem to be too advanced and out of my league for the moment.

So, i would very much appreciate if any of you could direct me to a source where i could find undergraduate articles in topology.

Thanks for your time and help!

 

[owlpride]

Point-set topology is pretty much a dead field research-wise. There are other topology topics though that are accessible without much background. Knot and graph theory come to mind as popular topics for undergraduate research projects. Have you expressed an interest in research to your professors?

 

[sutupidmath]

Point-set topology is pretty much a dead field research-wise.

This is indeed the feeling that i got after considerable hours browsing the internet in an attemt to find a suitable research topic/article.

Have you expressed an interest in research to your professors?

Yes indeed! I have already talked to one professor. But the main problem is my lack of background in Topology. This semester is going to be my first exposure to topology ( we are using Topology by Munkres), so it looks difficult to find something that i could start working rightaway. All the articles that i have come across so far seem to include topics that we are going to get to by the end of the semester.

 

[mrb]

What made you decide you wanted to do research in topology before knowing any?

Point set topology is definitely not a hot topic of research these days, but it's not quite true that it's completely dead - a professor at my undergrad school published in point set topology.

 

[sutupidmath]

What made you decide you wanted to do research in topology before knowing any?

First, when i was introduced to some basic topology of R, in my real analysis class, i liked that part the most in the whole course. So, i decided to take Topology with honors this semester, and thought i could try to do some research as part of it. But most probbably i am going to end up doing something else...we (my prof. and i) are still exploring some other options/projects that would be interesting.

Maybe i overemphasized a little bit when i said that i want to do research. More precisely, what i am looking for, at least for now, is a project that i could work on for the rest of the semester (maybe after all this is kind of research!)

Point set topology is definitely not a hot topic of research these days, but it's not quite true that it's completely dead - a professor at my undergrad school published in point set topology.

Correct! I also found a few articles in point-set topology, but the downfall of them, like i previously said, seems to be that all of them(the ones that i have seen) use concepts like Local onnectedness, Metric Topology, Local Compactness, Separation Axioms, Normal Spaces etc., topics that we won't get to until the second part of the semester. So, most probbably i will have to postpone it until next semester or later on.

"Semi-Open Sets and Semi-Continuity in Topological Spaces" by Norman Levine, was maybe the most accessible article to me.

 

[owlpride]

Point-set topology is typically considered a tool rather than an end in and of itself. The important and easy stuff has been taken care of (half) a century ago. For example, the article by Levine was published in 1963.

You might have a hard time finding an accessible research topic in an area you have no background in. I applaud your initiative to talk to your professors. I hope the two of you can find something you are interested in.

If you cannot locate an interesting project, have you considered using the time to just read about some topic in topology instead? You could get a head start on the theory of manifolds, for example. A couple of friends of mine have taken courses in algebraic and/or differential topology without a course in point-set topology. It is definitely doable, though it might be hard without the guidance of a professor. Or you could look at The Knot Book by Colin Adams. He starts with elementary knot theory and introduces some sophisticated topology towards the end (e.g. Dehn Surgery - the construction of three-dimensional manifolds via cutting and pasting along links - and the Poincare Conjecture).

If you like the "clean and sterile" feeling of point-set topological arguments more than the geometric intuition behind them, you could also take a look at logic and set theory.