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By Giovanni Landi (auth.)

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Extra info for An Introduction to Noncommutative Spaces and their Geometries: Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment

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If x ≺ y and there is no z such that x ≺ z ≺ y, then x is at the level immediately below y and these two points are connected by a link. 3) and for P4 (S 1 ). For the former, the partial order reads α ≺ a, α ≺ c, β ≺ a, β ≺ b, γ ≺ b, γ ≺ c. The latter is a four point approximation of S 1 obtained from a covering consisting of two intersecting open sets. The partial order reads x1 ≺ x3 , x1 ≺ x4 , x2 ≺ x3 , x2 ≺ x4 . In Fig. 1, (and in general, in any Hasse diagram) the smallest open set containing any point x consists of all points which are below the given one, x, and can be connected to it by a series of links.

The proof consists in constructing explicitly the Bratteli diagram D(A) of the algebra A. We shall sketch the main steps while referring to [18, 19] for more details. • Let {K0 , K1 , K2 , . } be the collection of all closed sets in the lattice P , with K0 = P . • Consider the subcollection Kn = {K0 , K1 , . . , Kn } and let Kn be the smallest collection of (closed) sets in P containing Kn which is closed under union and intersection. • Consider the algebra of sets13 generated by the collection Kn .

1 that each U is already a subtopology, namely that U = τ (U). In Sect. 1 we have associated with each covering Ui a T0 -topological space Pi and a continuous surjection π i : M → Pi . 24) defined whenever i ≤ j and such that πi = πij ◦ πj . 25) These maps are uniquely defined by the fact that the spaces Pi ’s are T0 and that the map πi is continuous with respect to τ (Uj ) whenever i ≤ j. Indeed, (−1) if U is open in Pi , then πi (U ) is open in the Ui -topology by definition, thus it is also open in the finer Uj -topology and πi is continuous in τ (Uj ).

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