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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),

[tex]\xi \equiv \sigma \,\text{on}\, \Lambda \Leftrightarrow \forall x\in\Lambda ,\, \xi(x) = \sigma(x)[/tex]

We defined the cylinder sets as

[tex]\mathcal{N}_\Lambda(\sigma) = \left\{ \xi\in\Omega : \xi \equiv \sigma \,\text{on}\, \Lambda\right\},\,\sigma\in\Omega[/tex]

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

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# Cylinder sets, some clarification

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