Cylinder sets, some clarification

[JorisL]

Hey,

In a course were we are treating phase transitions from a mathematically exact point of view, Cylinder sets were introduced. I'll first outline the context some more.

So we are considering systems on a lattice with a finite state space for each lattice point, for simplicity. E.g. an Ising spin model on a d-dimensional lattice $\mathcal{L} = \mathbb{Z}^d$.
The spins can take values in $S = \{-1,+1\}$ and the entire configuration is an element of $\Omega = S^\mathcal{L}$.
From Tychonoff's theorem we get that the configuration space over the entire lattice is compact (product of compact spaces).

Now denote $\Lambda \subset \mathcal{L}$ as a finite subset (for example the d-dimensional hypercubes $\Lambda_n = [-n,n]^d$).
Furthermore we use $\sigma , \nu , \xi$ to denote configurations on the lattice i.e. they are exactly the elements of $\Omega$.
Finally we have this notion we can use to have an idea of distance (not very exact I think),
$$\xi \equiv \sigma \,\text{on}\, \Lambda \Leftrightarrow \forall x\in\Lambda ,\, \xi(x) = \sigma(x)$$

We defined the cylinder sets as
$$\mathcal{N}_\Lambda(\sigma) = \left\{ \xi\in\Omega : \xi \equiv \sigma \,\text{on}\, \Lambda\right\},\,\sigma\in\Omega$$

I hope this is somewhat clear so far.
Mathematically it kind of makes sense to me, what I don't have is an intuitive idea of these cylinder sets.

How useful are these without the notion of distance (at least we haven't defined this so I guess it is not needed)?
Why are they called cylinder sets? Were they first constructed for a space where they actually are cylinders (or are isomorphic to cylinders)?
Can they be represented in a diagram (with appropriate projections/maps between space) like in the case of a manifold with its tangent spaces/bundle and cotangent spaces? I ask this because I attempted something like that and got stuck with making this into a clear diagram (other spaces are added afterwards like a Banach space of observables).

Thanks

Joris

 

[wabbit]

Your exposition is very clear - my take on this is that a cylinder is the set of points that project onto a given set (say a circle). In your case, you have a natural projection from the set of functions on the lattice to the set of functions on the subset of the lattice (the projection is of course just the restriction operation). As you can see, your cylinder set is indeed "what projects to something given".
I hope it's clear despite the lack of formulas : )