Theory of Ordered Spaces¶II. The Local Differential Structure

Abstract:

In this paper we investigate the conditions under which the ordered spaces defined in [1] are locally diffeomorphic to ℝ^N. In Sect.~1 we give an introduction and an overview of the results. In Sect. 2 we show that the axioms of [1] do not suffice to make light rays locally homeomorphic to ℝ. We introduce this structure via the new connectedness axiom 2.13, and work out some of its immediate consequences. In Sect. 3 we give the (somewhat involved) construction of timelike curves in a D-set, which are basic to everything that follows. They are used in Sect. 4 to prove (i) a nested interval theorem for ordered spaces; (ii) the contractibility of order intervals in D-sets; and (iii) that order intervals in D-sets are star-shaped. The notion of D-countability (meaning that a D-set has a countable base in the subspace topology) is introduced in Sect. 5. The Urysohn lemma shows that a D-countable ordered space is locally metrizable. If this space is also locally compact, then it has finite topological dimension N; these results are established in Sect. 6. The local differential structure now follows from known results: the embedding of such spaces in ℝ^{2 n +1}, and the result that an open star-shaped region in ℝⁿ is diffeomorphic to ℝⁿ. In conclusion, we exhibit these inclusions in Fig. 3, and suggest the possibility that Wigner's position on the “Unreasonable effectiveness of mathematics in the natural sciences” may be open to reasonable doubt. The axioms of [1] are given in the Appendix.