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Sorry if this deviates from the norm a bit, but I need to come up with a topological space to write a report on in these last couple of weeks now that all of my other assignments are out of the way. I have such little understanding of the actual application of this topic outside of standard metric spaces and subspaces, so I was hoping someone might have an interesting idea related to my interests.

It's important to note that I initially was studying chemistry three years ago before switching to math because I absolutely loved linear algebra and elementary differential equations. I've taken the graduate level courses in both of those subjects (and PDEs) already... I'm not sure if that will be entirely helpful for what I need to do, but I'm not afraid to do something that might require more than a surface level understanding of those subjects. Also, I'm open to things that might apply to chemistry as well!

Algebraic
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    Try the line with two origins topology...google it. – DonAntonio Nov 26 '20 at 05:53
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    Most of the fun stuff doesn't really happen until you learn at least a little bit of algebraic topology, so you can compute fun invariants like the fundamental group and (co)homology. You may be interested in learning about finite topological spaces: http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf – Qiaochu Yuan Nov 26 '20 at 06:35
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    The standard reference for fun topological spaces is Steen and Seebach Counterexamples in Topology. Connectedness properties are fairly intuitive and there are a number of odd examples involving those properties, such as the topologist's sine curve; the deleted comb space; the long line. – MJD Nov 26 '20 at 08:40
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    Relevant: What application is there for a non-Hausdorff topological space? – MJD Nov 26 '20 at 08:44
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    I found the Counterexamples in Topology on Scribd just now - great recommendation. I'll have to check out your link in a little bit. – Algebraic Nov 26 '20 at 09:04
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    @MJD That post answers questions I didn't even know I had. Thank you so much! – Algebraic Nov 26 '20 at 09:23
  • I'm glad I could help. – MJD Nov 26 '20 at 09:28
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    Other interesting spaces are the Sorgenfrey Line (a.k.a. the lower-limit topology on $\Bbb R$ ) and the Niemitzky (Moore-Niemitzky) plane. And you can look into Hilbert space(s), which have applications in quantum mechanics & in many more subjects. – DanielWainfleet Nov 26 '20 at 10:32

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Topology is a rather broad field, so there are very many interesting spaces to cover.

My personal favorite is the Sierpinski-space, which is the space $\{\bullet, \circ\}$, with precisely one nontrivial open set $\{\circ\}$. It is interesting out of the following reasons:

  • It is the smallest nontrivial topological space
  • It can be used to detect or classify open/closed subspaces of any topological space $X$
  • It is the Zariski-Topology of any arbitrary local integral domain
  • Possibly even more interesting facts

If you can pick a whole class of spaces I would suggest considering counterexamples for connectivity assumptions, by which I mean spaces like the Warshaw circle, Hawaiian earrings, topologists comb. They are interesting in the way how rather small modifications to a construction can fundamentally change topological properties.

These are examples rooted in abstract mathematics though and I would love to read about interesting spaces arising in applications...

Jonas Linssen
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  • Yeah, "tangible" examples have always been my favorite, but I really like your recommendation. I was considering knots and the cantor set, but someone else is already doing knots, and I literally took a course on chaotic systems only a few months ago.

    I'm surprised I didn't read about this in Crossley's book. They must have nearly 100 worked examples in the 200 page "Essential Topology", but nothing on this seemingly fundamental example.

     – Algebraic Nov 26 '20 at 08:11
  • Also, thanks for mentioning the counterexamples. That is certainly going to be a section of the paper. – Algebraic Nov 26 '20 at 08:14