This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological spaceView 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
Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

compactly generated space 




Opposite properties