### On equivalence relations on a differential space

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The most elegant definition of singularities in general relativity as b-boundary points, when applied to the closed Friedman world model, leads to the disastrous situation: both the initial and final singularities form the single point of the b-boundary which is not Hausdorff separated from the rest of space-time. We apply Alain Connes' method of non-commutative geometry, defined in terms of a C*-algebra, to this case. It turns out that both the initial and final singularities can be analysed as...

[For the entire collection see Zbl 0742.00067.]Differential spaces, whose theory was initiated by R. Sikorski in the sixties, provide an abstract setting for differential geometry. In this paper the author studies the wedge sum of such spaces and deduces some basic results concerning this construction.

The Legendre transformations on differential spaces (in the sense of Sikorski) is studied, and some properties for spaces with singularities are investigated. A mechanical interpretation of the Legendre transformation is also given.

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