The Lorentzian splitting theorem without the completeness assumption

Fundamental Theorem of Geometry without the 1-to-1 Assumption

It is proved that any mapping of an n-dimensional affine space over a division ring D onto itself which maps every line into a line is semiaffine, if n ∈ {2, 3, . . . } and D 6= Z2. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also –...

B -AND B - COMPLETENESS IN LOCALLY CONVEX ALGEBRAS AND THE E x THEOREM

Let E be a B-complete (B -complete) locally convex algebra and $ the topological direct sum of countably many copies of the scalar field with a jointly continuous algebra multiplication. It has been shown that E is also B-complete (B -complete) for componentwise multiplication on E . B-and Br-completeness of E , the unitization of E, and also of E x for other multiplications on E ...

The Splitting Theorem for Orbifolds

Achronal Limits, Lorentzian Spheres, and Splitting

In the early 1980s Yau posed the problem of establishing the rigidity of the Hawking–Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets—the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case....

The Gödel Completeness Theorem for Uncountable Languages1

This article is the second in a series of two Mizar articles constituting a formal proof of the Gödel Completeness theorem [15] for uncountably large languages. We follow the proof given in [16]. The present article contains the techniques required to expand a theory such that the expanded theory contains witnesses and is negation faithful. Then the completeness theorem follows immediately.