Tim Porter finite T0 spaces

Finite $T_0$ spaces

Definition

A topological space $X$ is a $T_0$ space if given distinct points of $X$, there is an open set of $X$ that contains one but not the other.

Definition

A $T_0$-space gives rise naturally to a partial order? on the set of points of $X$, where $x\leq y$ if for each open set, $U$, of $X$, $y\in U$ implies $x\in U$ and conversely.

Finite spaces from finite observations

The Sorkin model, (Sorkin:1991).

Let $X$ be a space, (which we are about to ‘observe’ because we want to ‘study’ it) and $\mathcal{F}$, a (locally finite) open cover of $X$. (The idea of the model is approximately that open sets correspond to observations, so if $x, x^\prime \in U$, the observation, $U$, ‘tests positive’ on both $x$ and $x^\prime$, so does not distinguish them.)

Using $\mathcal{F}$, define, on the set $X$, an equivalence relation $\sim_\mathcal{F}$, given by

• $x\sim_\mathcal{F} x^\prime$ if and only if, for all $U \in \mathcal{F}$, $x \in U\Leftrightarrow x^\prime \in U$\enspace ,

thus two points of $X$ are equivalent if all the observations from $\mathcal{F}$ give the same positive or negative result on them both. Using $\sim_\mathcal{F}$, we can form a quotient space, $X_\mathcal{F}$.

Gloss

1. In some ways, this seems silly, since as we do not know $X$, we do not know its topology and so should have little or no knowledge of the quotient topology on $X_\mathcal{F}$. The point is, however, that $X_\mathcal{F}$ is something we do know. It encodes the observational data on the mysterious (and perhaps ‘pointless’), $X$. The type of simplified model of ‘observational data’, using an idea that ‘observations behave like open sets’, does determine the model, but the type of construction is almost generic. The space $X_\mathcal{F}$ organises the data.

2. The question of the topology on $X_\mathcal{F}$ initially does look tricky, but quite generally it will be a $T_0$-space and hence will correspond to a partially ordered set in a natural way. The order can be specified without knowing the topology on $X$, merely needing the cover $\mathcal{F}$!

Writing $[x]_\mathcal{F}$ for the equivalence class of $x\in X$, in a very natural way:

• $[x]_\mathcal{F}\leq [x^\prime]_\mathcal{F}$ if and only if, for every open set, $U$ in $\mathcal{F}$, if $x^\prime\in U$, then $x \in U$.

In fact, in situations , in which the cover is finite, $X_\mathcal{F}$ is a finite $T_0$-space, and each point $[x]_\mathcal{F}$ is in a unique minimal open set, $U_{[x]}$ of $X_\mathcal{F}$, and

• $[x]_\mathcal{F}\leq [x^\prime]_\mathcal{F}$ if and only if $x \in U_{[x^\prime]}$

Of course, this has a nice interpretation in terms of ‘observations’. The essential information on $X_\mathcal{F}$ is contained in this partially ordered set and it can be considered to be a spatial representation of the original ‘space’ $X$, relative to the observations considered.