Why study finite topological spaces?

Question

In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:

… this means that some concepts that I use freely and naturally in my personal thinking are foreign to most mathematicians I talk to. My personal mental models and structures are similar in character to the kinds of models groups of mathematicians share—but they are often different models. At the time of the formulation of the geometrization conjecture, my understanding of hyperbolic geometry was a good example. A random continuing example is an understanding of finite topological spaces, an oddball topic that can lend good insight to a variety of questions but that is generally not worth developing in any one case because there are standard circumlocutions that avoid it.

When reading his papers one very quickly sees what he means about how unique his way of thinking about hyperbolic geometry was. I'm more intrigued by the remark about finite topological spaces: I'm aware they can yield quick counterexamples to a few naïve conjectures in general/algebraic topology. They are also clearly related to the theory of lattices, which are extremely useful.

But aside from the Sierpiński space (whose importance I appreciate the reasons for more easily) and the examples above how have finite topological spaces been used to yield insights into other parts of mathematics? And what would be some instances where ‘standard circumlocutions’ were used to avoid them?

Answer 1

Prompted by the same excerpt, I asked Bill Thurston in 2011:

Can you point me somewhere where I can read about some of your mental models and structures as they relate to finite topological spaces?

His response was as follows:

As to finite topology: I don't really know a good source to read about it. However, it's standard in combinatorics to talk about the topology associated with a partially ordered set (poset), which is really the same thing: from a finite order, you get a partial order by saying $x > y$ if the closure of the point $x$ contains the point $y$, and vice versa, from a partially ordered set, you get a topology whose open sets are the order ideals $U_y = \{ x \mid x \ge y \}$. Often people go straight from that to a kind of realization, analogous to the realization of a simplicial complex or simplicial set --- the realization of a finite poset can be defined recursively by (a) the realization of a closed point is a point, and (b) the realization of the smallest closed set $V_x$ containing $x$ is the cone on the realization of $V_x \smallsetminus {x}$. You can associate with each point in the finite topology the corresponding open cone in its realization; the finite topology is then the quotient space of its realization obtained by collapsing each of these contractible open cones to a point. As a consequence, the finite topology has the same weak homotopy type as its realization.

Answer 2

Long comment: Note that a finite topological space is nothing but a finite pre-order (order, iff the topology is $T_0$), given by $x\leq y$ iff $x\in \overline{\{y\}}$. An equivalence relation on a topological space, e.g. by considering the orbits of a group action, gives rise to a quotient space, which one endows naturally with the quotient topology, which is equivalent to a pre-order in the case where there are finitely many orbits. A good example to keep in mind is the action of a Borel subgroup $B$ on the coset space $G/B$, where $G$ is a semisimple group. Then the quotient space could be identified with the corresponding Weyl group, by Bruhat decomposition, which is thus endowed with an order, that is the Bruhat order.

Answer 3

As Uri Bader's answer notes, finite $T_0$ topological spaces are equivalent to finite partially ordered sets (posets). Now, combinatorialists who are interested in the topology of finite posets most commonly study the simplicial complex formed by all totally ordered subsets. While these two topologies on a poset are not exactly the same, McCord (Singular homology groups and homotopy groups of finite topological spaces) showed that there is a weak homotopy equivalence between the two. Therefore the study of the homotopy and homology of finite posets, which comprise a large class of finite simplicial complexes, is really the study of finite $T_0$ topological spaces.

For more information, see Björner's chapter on Topological Methods in Combinatorics in the Handbook of Combinatorics.

Answer 4

Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa: Are finite spaces a model for finite CW-complexes?

So, when it comes to computing homotopy groups of finitely triangulated spaces, there is really no loss of generality.

One might try to draw as an exercise a finite topological space having the same homotopy groups as the 2-sphere: it will be a set of cardinality $14$, namely the poset (order given by inclusion) of all simplices of the boundary of the $3$-simplex with the order topology.

Answer 5

And what would be some instances where 'standard circumlocutions' used to avoid them?

For example, one can define contractability and compactness (for metrizable spaces) in terms of maps of finite topological spaces and orthogonality of morphisms (see examples in the section on topological spaces in Ncatlab, Lifting property. Perhaps words of Thurston apply to these examples: these reformulations might "lend good insight to a variety of questions" in topology "but that is generally not worth developing in [this] case because there are standard circumlocutions that avoid it", namely these standard circumlocutions are the standard theory of contractability and compactness in any textbook.

Answer 6

This question came to mind upon M W's answer to this recent Math.SE question - the question asked about finite spaces, but it was pointed out that all that was really needed was the Alexandrov property, the class of spaces for which the intersection of an arbitrary collection of open sets is open.