By Giovanni Landi (auth.)
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Additional info for An Introduction to Noncommutative Spaces and their Geometries: Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment
Example text
Proposition 12. The image π∞ (M ) is dense in P∞ . Proof. If U ⊂ P∞ is any nonempty open set, by the definition of the topology (−1) of P∞ , U is the union of sets of the form πi∞ (Ui ), with Ui open in Pi . Choose xi ∈ Ui . Since πi is surjective, there is at least a point m ∈ M , for which πi (m) = xi and let π∞ (m) = x. Then πi∞ (x) = πi∞ (π∞ )(m) = πi (m) = xi , (−1) (−1) from which x ∈ πi∞ (xi ) ⊂ πi∞ (Ui ) ⊂ U . This proves that π∞ (M )∩U = ∅, which establishes that π∞ (M ) is dense. Proposition 13.
In this Chapter we shall describe noncommutative lattices in some detail while in Chap. 11 we shall illustrate some of their applications in physics. 1 The Topological Approximation The approximation scheme we are going to describe has really a deep physical flavour. To get a taste of the general situation, let us consider the following simple example. Let us suppose we are about to measure the position of a particle which moves on a circle, of radius one say, S 1 = {0 ≤ ϕ ≤ 2π, mod 2π}. Our ‘detectors’ will be taken to be (possibly overlapping) open subsets of S 1 with some mechanism which switches on the detector when the particle is in the corresponding open set.
1. The Hasse diagrams for P6 (S 1 ) and for P4 (S 1 ) Ui+1 Ui−1 ... ( ) ) ( ( ) Ui ) ( ( Ui+2 ) ... π ❄ yi−2 ··· ❅ s ❅ yi−1 s ❅ ❅ xi−2 yi s ❅ s ❅ xi−1 yi+1 s ❅ ❅ s ❅ xi ❅ s ❅ s ❅ ❅ xi+1 s ❅ ··· xi+2 Fig. 2. The finitary poset of the line R The generic finitary poset P (R) associated with the real line R is shown in Fig. 2. The corresponding projection π : R → P (R) is given by Ui+1 \ {Ui ∩ Ui+1 Ui ∩ Ui+1 −→ xi , i ∈ Z , Ui+1 ∩ Ui+2 } −→ yi , i ∈ Z . 17) A basis for the quotient topology is provided by the collection of all open sets of the form Λ(xi ) = {xi } , Λ(yi ) = {xi , yi , xi+1 } , i ∈ Z .