By Giovanni Landi

An advent to a number of rules & functions of noncommutative geometry. It starts off with a no longer inevitably commutative yet associative algebra that's regarded as the algebra of features on a few digital noncommutative area.

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72) which, in turn, implies that the mapping F → ΛF ↔ IF is injective. To show surjectivity, let I be an ideal in A with associated subdiagram ΛI . For n = 0, 1, . , deﬁne Fn = P \ {Yn (k) | ∃(n − 1, p) ∈ ΛI , (n − 1, p) 1 (n, k) ∈ ΛI } . 73) k Then {Fn }n is a decreasing sequence of closed sets in P . By assumption (iv), there exists an m such that Fm = n Fn . By deﬁning F = Fm , one has Fn = F for n ≥ m and ΛI {(n, k) | n ≥ m} =: (ΛF )m . 74) Thus, ΛI = ΛF and the mapping F → IF is surjective.

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. Let M be T0 and the collection {Ui } of coverings be such that for every m ∈ M and every neighborhood N m, there exists an index 32 3 Projective Systems of Noncommutative Lattices π∞ M ❆❅ ❆❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ ❆ ❅ πj ❆ ❅ ❆ ❅ ❆ ❘ ❅ ❆ πi ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✲ P∞ ✁ ✁ ..

For example, one can prove that for the ﬁrst homotopy group, π1 (PN (S 1 )) = Z = π(S 1 ) whenever N ≥ 4 [141]. Consider the case N = 4. Elements of π1 (P4 (S 1 )) are homotopy classes of continuous maps σ : [0, 1] → P4 (S 1 ), such that σ(0) = σ(1). With a any real number in the open interval ]0, 1[, consider the map x3 if t = 0 x2 if 0 < t < a σ(t) = x4 if t = a . 4 shows this map for a = 1/2; the map can be seen to ‘wind once around’ P4 (S 1 ). 20) is manifestly continuous, being constructed in such a way that closed (respectively open) points of P4 (S 1 ) are the images of closed (respectively open) sets of the interval [0, 1].