Tim Porter Finite Topological Spaces

Idea

This is a project to explore the (directed) homotopy theory of finite topological spaces.

References

• Jonathan Barmak, Topología Algebraica de Espacios Topológicos Finitos y Aplicaciones

• Homotopy Theory of Finite Topological Spaces

• INVERSE LIMITS OF FINITE TOPOLOGICAL SPACES

• Finite Topological Spaces: May

Discussion

October 20, 2010

Eric: Are we going to

explore the directed homotopy theory of finite topological spaces

or

explore the directed homotopy theory of directed finite topological spaces

???

All finite spaces are directed in some way (being posets), but in the latter sense, I mean to make contact with directed spaces (Grandis), concurrent systems, etc.

If it is the former, that is still interesting, but then I will stop blabbering about “speed of information”, diamonds, etc.

Personally, I think it would be fun to develop a theory of finite directed spaces and study the directed homotopy (among other things) of those spaces.

PS: I see that Barmak introduces a continuum interval when defining homotopy in a clever way, but it just begs the question, “Why bother?” He is working with finite spaces already, why not go “full finite” and look at homotopy using a finite version of the interval?

Lemma 1.2.3. Let $x$, $y$ be two comparable points of a finite space $X$. Then, there exists a path from $x$ to $y$ in $X$, i.e. a map $\alpha$ from the unit interval $I$ to $X$ such that $\alpha(0) = x$ and $\alpha(1) = y$.

Proof. Assume $x\le y$ and define $\alpha : I \to X$, $\alpha(t) = x$ if $0 \le t \lt 1$, $\alpha(1) = y$. If $U \subseteq X$ is open and contains $y$, then it contains $x$ also. Therefore $\alpha^{-1}(U)$ is one of the following sets, $\emptyset$, $I$ or $[0, 1)$, which are all open in $I$. Thus, $\alpha$ is a continuous path from $x$ to $y$.

October 19, 2010

Tim Slightly off topic but interesting I think are two areas that seem distantly related to this one.

1. Causal site (Christensen and Crane 2005)

2. The logical/CS/AI area called Qualitative Spatial Reasoning (http://www.comp.leeds.ac.uk/qsr/) which aims to look at the logic of interacting regions.

I will try an type up some stuff on causal sites to start with. That is based, … yes!, on diamonds!Discrete Causal Spaces

October 18, 2010

Eric: I suppose we should start with some definitions? Settling notation, etc :)

At a very simple level, I think of finite topological spaces as beginning with some finite set, e.g. $\{a,b,c\}$. We can put a topology on this by declaring a set of subsets to be “closed” if the set contains all subsets of the biggest set.

There is probably a better way to say that :)

For example,

$$\Delta_{a,b,c} = \{\{a,b,c\},\{a,b\},\{b,c\},\{a,c\},\{a\},\{b\},\{c\},\{\}\}$$

is closed. Similarly,

$$\Delta_{a,b} = \{\{a,b\},\{a\},\{b\},\{\}\}$$

and

$$\Delta_{b,c} = \{\{b,c\},\{b\},\{c\},\{\}\}$$

are closed.

The interior of $\Delta_{a,b,c}$ is open, i.e.

$$int(\Delta_{a,b,c}) = \{\{a,b,c\}\}.$$

Then we get familiar identities from topology on $\mathbb{R}^n$ such as

$$\Delta_{a,b,c} \cap \Delta_{a,b} = \Delta_{a,b},$$

${}$

$$\Delta_{a,b} \cap \Delta_{b,c} = \Delta_{b},$$

etc.

However, this topological space is obviously non-Hausdorff (since the topology is not discrete).

Tim In fact, Hausdorff$\Rightarrow$Uninteresting!

Eric: Yes! Which itself is interesting :)

By the way, I expect a finite directed space to be quite a different beast than a finite space. So I think introducing finite spaces is just a warm up. For example, an (undirected) closed finite 1-simplex would be the set (open to notation suggestions!)

$$\Delta_{a,b} = \{\{a,b\},\{a\},\{b\},\{\}\}.$$

However, a directed 1-simplex would probably be something like

$$\uparr\Delta_{a,b} = \{\{a,b\},\{a\},\{\}\}$$

or something.

In other words, I don’t think a finite directed space should be a finite space with something thrown on top, but instead, something fundamentally of a different nature.

But I’m just thinking out loud…

Tim: What are interesting are finite T0 spaces. They are naturally directed. (see the link).

Eric: I think I see (getting ahead of myself as I haven’t finished reading the basics). A finite directed set should be a simple path in a Hasse diagram such as the example I gave above:

$$\uparr\Delta_{a,b} = \{\{a,b\},\{a\},\{\}\}.$$

The undirected version

$$\Delta_{a,b} = \{\{a,b\},\{a\},\{b\},\{\}\}.$$

contains two paths in the Hasse diagram.

This probably means I do not want to include the empty set to make that perfectly clear, i.e.

$$\uparr\Delta_{a,b} = \{\{a,b\},\{a\}\}.$$

category: project