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I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the lower limit...). That is, I only know how to think about a topological space if it happens to live on one of these islands, the methods appropriate to one island may be completely unrelated to those of another, and a "random" topological space is probably unrelated to anything I know how to think about and is thus "pathological". I'd like to get a better perspective on how many of these islands there are -- and perhaps whether some which I think are distinct are actually connected by some isthmus. Here's what my current map looks like:

  • CW complexes (and spaces homotopy equivalent to such)

  • Zariski spectra of commutative rings (and schemes)

  • Stone spaces (totally disconnected compact Hausdorff spaces)

  • Infinite-dimensional topological vector spaces (and spaces locally modeled on them)

One of the characteristic features of this map is that there is little overlap between the islands although there is overlap between these islands in a literal sense, what really sets them apart is that the tools used in exploring one island bear little resemblance to those used for another. For example, when studying spaces using CW complex tools, non-Hausdorffness is regarded as pathological, clopen sets as uninteresting, and infinite-dimensionality as an annoyance whereas such features are respectively embraced when studying Zariski spectra, Stone spaces, and functional-analytic spaces.

Questions:

  1. Are there other classes of topological spaces which are interesting to study (and not just as a source of pathologies)?

  2. Are these islands less isolated than I'm making them out to be? E.g. are there interesting topological considerations to be made which apply simultaneously to, say, Banach spaces and Stone spaces?

  3. Is it correct to think that the ocean is vast, i.e. that "most" topological spaces are "pathological"?

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    $\begingroup$ Stone spaces are Zariski spectra of Boolean rings. $\endgroup$ – Todd Trimble Jun 6 at 5:00
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    $\begingroup$ Polish spaces as another class? $\endgroup$ – Todd Trimble Jun 6 at 5:17
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    $\begingroup$ Another vote for Polish spaces from me. They're the typical "nice" space for measure theory, probability, and descriptive set theory. Elsewhere in analysis, "locally compact Hausdorff" is an important island. $\endgroup$ – Nate Eldredge Jun 6 at 6:32
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    $\begingroup$ The sea of pathology has various levels of depth: there is the deep sea of non-$T_0$ spaces, but there are deeper trenches: pretopological spaces and pseudotopological spaces. Of what unseen horrors in these depths may lurk few have lived to tell the tale. Also, let's not forget the great ocean of locales (connected by the strait of sober spaces) in which perhaps more islands of non-pathology can be found. $\endgroup$ – Gro-Tsen Jun 6 at 12:00
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    $\begingroup$ I find it difficult to see how $\omega_1+1$ with its order topology is on an island (of Stone spaces) whereas inserting pieces of $\mathbb R$ to make the long line pushes the space into the ocean. $\endgroup$ – Andreas Blass Jun 6 at 17:01
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What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.

In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.

Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $\mathbb{R}^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.

(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)

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    $\begingroup$ Some of the 'islands' are very close to each other, so the analogy of a map has its limitations. Any compact metric (CM) space can be thought of as sitting in the Hilbert cube, and then it can be thought of as an intersection of polyhedral neighbourhoods so although CM spaces can be pathological from some points of view they are sort of infinitely near the polyhedral space island!!! It would be fun to try to extend the map with bridges between islands, (perhaps adjoint functors might serve for this) and then talk of reefs that are almost above sea level, etc. $\endgroup$ – Tim Porter Jun 8 at 14:02
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I'll go ahead and say that Polish spaces are an interesting and almost sui generis class. There is a rich literature of applications to and from descriptive set theory, with layers of "pathology" hierarchically organized along lines closely related to the arithmetical and analytic hierarchies. They are also widely used, as Nate mentioned, in abstract measure and probability theory.

I would say there are isthmuses connecting this class to the class of continua (mentioned by D.S. Lipham; consider for example the theory of pseudo-arcs) as compact connected metric spaces, and to some extent to locally compact Hausdorff spaces (e.g., a locally compact Hausdorff space is Polish iff it is second countable).

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o-minimal structures, as generalisation of “tame” topology of semi-algebraic and semi-analytic sets, are quite interesting, in applications in particular. See e.g. the book

https://books.google.co.uk/books/about/Tame_Topology_and_O_minimal_Structures.html

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  • $\begingroup$ A great book which deserves to be well-known. $\endgroup$ – Todd Trimble Jun 8 at 11:49

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