Alexandrov space

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This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space is termed an Alexandrov space or finitely generated space if it satisfies the following equivalent conditions:

No. Shorthand A topological space is termed Alexandrov (or finitely generated) if ...
1 arbitrary intersections of open subsets any arbitrary intersection of open subsets (including intersections of infinitely many open subsets) is open
2 arbitrary unions of closed subsets any arbitrary union of closed subsets (including unions of infinitely many closed subsets) is closed
3 unique smallest open subset containing any point any point has a unique smallest open subset containing it.
4 complements give another topology there is another topology on the set where the closed subsets are precisely the open subsets of this topology, and the open subsets are precisely the closed subsets of this topology.
5 generated by finite subsets the topology is generated by finite subsets, i.e., a subset of the space is open iff its intersection with every finite subset is open in the subspace topology on that subset.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite space |FULL LIST, MORE INFO
discrete space Almost discrete space|FULL LIST, MORE INFO
locally finite space every point is contained in a finite open subset |FULL LIST, MORE INFO
almost discrete space Alexandrov and zero-dimensional |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
compactly generated space

Opposite properties