Alexandrov topology

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Alexandrov topology

Wikipedia

Etymology

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

Noun

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) ${\displaystyle S_{0,2}(P(R_{C}))}$ is the family of open sets of the Alexandrov topology induced by ${\displaystyle R_{C}}$ and ${\displaystyle S^{0,5,7}(P(R_{C}))}$ is the family of closed sets of the Alexandrov topology induced by ${\displaystyle R_{C}^{\smile }}$;

Usage notes

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.