Let be a set and the set of subsets of . A function is a closure operation if for all :
K5) . These are the Kuratowski closure axioms, and a function satisfying these axioms is called a Kuratowski closure operator (or Kuratowski closure operation).
A pair is a closure space if and only if satisfies:
This condition is equivalent to K1)–K3). A closure space that satisfies K5) is a Fréchet -space. A Fréchet -space that satisfies K4) is a topological space.
A Čech closure operator is a mapping such that
C2) ( K1)) ;
C3) . A set endowed with a Čech closure operator is called a pre-topological space.
|[a1]||N.M. Martin, S. Pollard, "Closure spaces and logic" , Kluwer Acad. Publ. (1996)|
|[a2]||J.L. Kelley, "General topology" , v. Nostrand–Reinhold (1955)|
|[a3]||D. Dikranjan, W. Tholin, "Categorical structures of closure operators" , Kluwer Acad. Publ. (1996)|
Closure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_space&oldid=13257