Closure space
From Encyclopedia of Mathematics
Let be a set and the set of subsets of . A function is a closure operation if for all :
K1) ;
K2) ;
K3) ;
K4) ;
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
C1) ;
C2) ( K1)) ;
C3) . A set endowed with a Čech closure operator is called a pre-topological space.
References
[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) |
How to Cite This Entry:
Closure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_space&oldid=13257
Closure space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_space&oldid=13257
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article