Alexandroff space

Definition
A topological space is termed a principal space or Alexandroff space if it satisfies the following equivalent conditions:


 * Every point has a unique minimal neighbourhood
 * An arbitrary union of open subsets is open
 * An arbitrary union of closed subsets is closed

Stronger properties

 * Trivial space (viz a set with the trivial topology)
 * Discrete space

Weaker properties

 * Orthocompact space