Theory of ordered spaces II. The local differential structure

H. J. Borchers, R. N. Sen

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


In this paper we investigate the conditions under which the ordered spaces defined in [1] are locally diffeomorphic to ℝN. 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 ℝ2N+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.

Original languageEnglish
Pages (from-to)475-492
Number of pages18
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - 1 Jan 1999

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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