Alexandrov topology

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English

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Etymology

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Along with the closely related concept of Alexandrov-discrete space, after Russian mathematician Pavel Sergeyevich Alexandrov.

Noun

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Alexandrov topology (plural Alexandrov topologies)

  1. (topology) A topology in which the intersection of any family of open sets is an open set.
    • 1972, Roger Penrose, Techniques of Differential Topology in Relativity, SIAM, page 34:
      4.24. THEOREM. The following three restrictions on a space-time M are equivalent:
      (a) M is strongly causal;
      (b) the Alexandrov topology agrees with the manifold topology;
      (c) the Alexandrov topology is Hausdorff.
    • 1990, Palle Yourgrau, Demonstratives, Oxford University Press, page 252:
      In Minkowski space-time there is, indeed, a topology definable solely in terms of the causal connectivity among events, the Alexandrov topology, which is provably equivalent to the ordinary manifold topology.
    • 2016, Piero Pagliani, Covering Rough Sets and Formal Topology — A Uniform Approach Through Intensional and Extensional Operators, James F. Peters, Andrzej Skowron (editors), Transactions on Rough Sets XX, Springer, LNCS 10020, page 134,
      (6) is the family of open sets of the Alexandrov topology induced by and is the family of closed sets of the Alexandrov topology induced by ;

Usage notes

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By the axioms of topology, the intersection of any finite family of open sets is open; in Alexandrov topologies the same is true for infinite families.