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**

**Sample text**

It is not even a T1 -space; again, in P6 (S 1 ) only the points a, b and c are closed while the points α, β and γ are open. In general there will be points which are neither closed nor open. It can be shown, however, that PU (M ) is always a T0 -space, being, indeed, the T0 -quotient of M with respect to the topology U [141]. 2 Order and Topology What we shall show next is how the topology of any ﬁnitary T0 topological space P can be given equivalently by means of a partial order which makes P a partially ordered set (or poset for short) [141].

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. The number of detectors must be clearly limited and we take them to consist of the following three open subsets whose union covers S 1 , G. Landi: LNPm 51, pp. 21–58, 2002. 1) U3 = {π < ϕ < 2π}. Now, if two detectors, U1 and U2 say, are on, we will know that the particle is in the intersection U1 ∩ U2 although we will be unable to distinguish any two points in this intersection.

Then the space PU (M ) will consist of a countable number of points; in the terminology of [141] PU (M ) would be a ﬁnitary approximation of M . 2) is P6 (S 1 ). 3) it is evident that in P6 (S 1 ), for instance, we cannot isolate the point a from α by using open sets. It is not even a T1 -space; again, in P6 (S 1 ) only the points a, b and c are closed while the points α, β and γ are open. In general there will be points which are neither closed nor open. It can be shown, however, that PU (M ) is always a T0 -space, being, indeed, the T0 -quotient of M with respect to the topology U [141].