I've lately been told that the subject of general topology, like what Hausdorff and Kolmogorov dealt with, is a a dead subject research wise. I have been wondering however whether this is indeed true, and would like to know whether there are people actively researching general topology or adjacent subjects? I would appreciate perhaps some concrete examples of people or institutions dealing with the subject.
Active research on the subject of general topology
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Related Solutions
Sundry advice on looking for ideas:
If you have not done so already, cultivate relationships with your faculty. If you think there is one that you would like to study with, they would probably be your best bet for a fast orientation to a topic you might pick. This is especially important if they can guide you on the type of work they're looking for in your thesis.
Go to all the math colloquia and talks that you can possibly attend. Chat with other students and professors about math topics whenever possible. Consider giving talks on what you have learned or what you are thinking about, also.
Read more papers on the things you like (not necessarily thoroughly, but just to get a feel for what you could learn about.) Just expose yourself to more stuff!
Don't let learning something new/hard put you off topics you like. In fact, there is even a possibility you might find it enjoyable to learn an entirely new thing to motivate your research. (Don't embark on that without deep reflection, though, if you are tight on time.)
Search the references of papers you like for other good things to check out. Make a binder. Highlight stuff.
If you find out about a mathematician who is working in the thing you like, ask around to see if they will bite your head off. If the consensus seems to be "no", work up the courage to strike up a conversation with them.
On $\mathbb R$ (or any other field, actually), the Zariski topology is exactly the cofinite topology, in which the closed sets are the finite ones. Example 18-19 of Steen and Seebach's "Counterexamples in topology" is dedicated to this.
One fact that you may find interesting is that every infinite subset is dense. An explicit example of this is that the sequence of positive integers converges to any point of $\mathbb R$. Indeed, if $x\in\mathbb R$ and $U$ is an open neighborhood of it, then the complement of $U$ is finite, hence it has a maximum $M$. If $n\in\mathbb N$ is sufficiently big, then $n>M$ and so $n\in U$. Thus $n\to x$.
In comments, Noah rightfully points out that in higher dimension the Zariski topology is more complicated than the cofinite one. On $\mathbb R^2$ with Zariski topology, for example, it is no longer the case that all infinite sets are dense. For example, the zero set of any polynomial is closed and a proper subset of $\mathbb R^2$, hence not a dense set.
However, any infinite set that is not contained in the zero set of a polynomial is dense. For example, $$ \{(x, y)\in\mathbb R^2 \ :\ y=\log\, \lvert x \rvert \}$$ is dense in $\mathbb R^2$.
Best Answer
Look in the library for a copy of Open Problems in Topology (two volumes so far), and Recent Progress in General Topology (2 volumes, last from 2002) and see that it is far from dead... Most of the problems in them are still open AFAIK. Elliott Pearl has some overview papers keeping track of which ones are solved or open IIRC.
The 1990 first book of Open Problems can be found here
A first status report is here
Publisher's link for the second volume of Open Problems is here
Second volume for Recent Progress: here
etc etc. The most relevant journal is "Topology and its Applications", and there is a journal just on questions called "Questions and Answers in General Topology" (though I rarely see it any more, it was Japanese, IIRC), and Fundamenta Mathematica has many general topology papers (mostly descriptive set theory and set-theoretic).
Many people think it's dead because it's not considered interesting, or "deep", but I disagree.