Alexandrov Topology - Characterizations of Alexandrov Topologies

Characterizations of Alexandrov Topologies

Alexandrov topologies have numerous characterizations. Let X = <X, T> be a topological space. Then the following are equivalent:

• Open and closed set characterizations:
  • Open set characterization. An arbitrary intersection of open sets in X is open.
  • Closed set characterization. An arbitrary union of closed sets in X is closed.

• Neighbourhood characterizations:
  • Smallest neighbourhood characterization. Every point of X has a smallest neighbourhood.
  • Neighbourhood filter characterization. The neighbourhood filter of every point in X is closed under arbitrary intersections.

• Interior and closure algebraic characterizations:
  • Interior operator characterization. The interior operator of X distributes over arbitrary intersections of subsets.
  • Closure operator characterization. The closure operator of X distributes over arbitrary unions of subsets.

• Preorder characterizations:
  • Specialization preorder characterization. T is the finest topology consistent with the specialization preorder of X i.e. the finest topology giving the preorder ≤ satisfying x ≤ y if and only if x is in the closure of {y} in X.
  • Open up-set characterization. There is a preorder ≤ such that the open sets of X are precisely those that are upwardly closed i.e. if x is in the set and x ≤ y then y is in the set. (This preorder will be precisely the specialization preorder.)
  • Closed down-set characterization. There is a preorder ≤ such that the closed sets of X are precisely those that are downwardly closed i.e. if x is in the set and y ≤ x then y is in the set. (This preorder will be precisely the specialization preorder.)
  • Upward interior characterization. A point x lies in the interior of a subset S of X if and only if there is a point y in S such that y ≤ x where ≤ is the specialization preorder i.e. y lies in the closure of {x}.
  • Downward closure characterization. A point x lies in the closure of a subset S of X if and only if there is a point y in S such that x ≤ y where ≤ is the specialization preorder i.e. x lies in the closure of {y}.

• Finite generation and category theoretic characterizations:
  • Finite closure characterization. A point x lies within the closure of a subset S of X if and only if there is a finite subset F of S such that x lies in the closure of F.
  • Finite subspace characterization. T is coherent with the finite subspaces of X.
  • Finite inclusion map characterization. The inclusion maps f_i : X_i → X of the finite subspaces of X form a final sink.
  • Finite generation characterization. X is finitely generated i.e. it is in the final hull of the finite spaces. (This means that there is a final sink f_i : X_i → X where each X_i is a finite topological space.)

Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician Pavel Alexandrov who first investigated them.