interior axioms
Let be a set. Then an interior operator is a function which satisfies the following properties:
Axiom 1.
Axiom 2.
For all , one has .
Axiom 3.
For all , one has .
Axiom 4.
For all , one has .
If is a topological space, then the operator which assigns to
each set its interior satisfies these axioms. Conversely, given an
interior operator on a set , the set defines a topology on in which is the
interior of for any subset of . Thus, specifying an
interior operator on a set is equivalent
to specifying a topology
on that set.
The concepts of interior operator and closure operator are closely
related.
Given an interior operator , one can
define a closure operator by the condition
and, given a closure operator , one can define an interior operator by the condition
Title | interior axioms |
---|---|
Canonical name | InteriorAxioms |
Date of creation | 2013-03-22 16:30:37 |
Last modified on | 2013-03-22 16:30:37 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 8 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 54A05 |
Related topic | GaloisConnection |
Defines | interior operator |