Tim Porter finite T0 spaces

Finite T 0T_0 spaces

Definition

A topological space XX is a T 0T_0 space if given distinct points of X X, there is an open set of XX that contains one but not the other.

Definition

A T 0T_0-space gives rise naturally to a partial order? on the set of points of XX, where xyx\leq y if for each open set, UU, of XX, yUy\in U implies xUx\in U and conversely.

Finite spaces from finite observations

The Sorkin model, (Sorkin:1991).

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

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

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

thus two points of XX 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 X_\mathcal{F}.

Gloss

  1. In some ways, this seems silly, since as we do not know XX, we do not know its topology and so should have little or no knowledge of the quotient topology on X X_\mathcal{F}. The point is, however, that X X_\mathcal{F} is something we do know. It encodes the observational data on the mysterious (and perhaps ‘pointless’), XX. 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 X_\mathcal{F} organises the data.

  2. The question of the topology on X X_\mathcal{F} initially does look tricky, but quite generally it will be a T 0T_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 XX, merely needing the cover \mathcal{F}!

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

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

In fact, in situations , in which the cover is finite, X X_\mathcal{F} is a finite T 0T_0-space, and each point [x] [x]_\mathcal{F} is in a unique minimal open set, U [x]U_{[x]} of X X_\mathcal{F}, and

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

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

Last revised on June 11, 2022 at 06:41:40. See the history of this page for a list of all contributions to it.