Undergraduate Research Topics in Point-Set Topology?

In summary, the conversation is about finding a suitable research topic in topology for an undergraduate student. The person is interested in point-set topology but is having difficulty finding accessible articles due to their lack of background. They have talked to a professor and are considering other topology topics such as knot and graph theory. The conversation also mentions the possibility of exploring manifolds, algebraic and differential topology, or logic and set theory as alternative research topics. However, the main focus is on finding a topic in point-set topology that is not too advanced for an undergraduate student.
  • #1
sutupidmath
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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!
 
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  • #2
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?
 
  • #3
owlpride said:
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.

owlpride said:
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.
 
  • #4
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.
 
  • #5
mrb said:
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!)
mrb said:
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.
 
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  • #6
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.
 
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1. What is point-set topology?

Point-set topology is a branch of mathematics that studies the properties and relationships of sets of points in a space, without considering any additional structure such as distance or angles. It is used to understand the topological properties of abstract spaces and provide a framework for understanding other areas of mathematics such as analysis, geometry, and algebra.

2. What are some common research topics in point-set topology for undergraduate students?

Some common research topics in point-set topology for undergraduate students include topological spaces, continuous functions, compactness, connectedness, separation axioms, and homotopy theory. Other topics may include metric spaces, topological manifolds, and algebraic topology.

3. How is point-set topology used in other fields?

Point-set topology is used in a variety of fields, including physics, computer science, and engineering. It is used to study the topology of physical spaces and objects, to develop algorithms and data structures in computer science, and to model and analyze complex systems in engineering.

4. What are some examples of real-world applications of point-set topology?

Point-set topology has a wide range of practical applications, including in GIS (geographic information systems), computer graphics, image processing, and medical imaging. It is also used in economics, sociology, and other social sciences to study networks and relationships between individuals or entities.

5. What skills and knowledge are necessary for undergraduate research in point-set topology?

To conduct research in point-set topology, students should have a strong foundation in mathematics, particularly in analysis and abstract algebra. They should also have a solid understanding of set theory, logic, and proof techniques. Familiarity with programming and computer software, such as Mathematica or MATLAB, may also be helpful for data analysis and visualization.

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