Preclosure operator
Encyclopedia
In topology
, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator
, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms
.
is a map 
where
is the power set of 
.
The preclosure operator has to satisfy the following properties:
The last axiom implies the following:
is closed (with respect to the preclosure) if 
. A set 
is open (with respect to the preclosure) if 
is closed. The collection of all open sets generated by the preclosure operator is a topology
.
The closure operator
cl on this topological space satisfies
for all 
.
a premetric on 
, then
is a preclosure on
.
is a preclosure operator. Given a topology 
with respect to which the sequential closure operator is defined, the topological space 
is a sequential space
if and only if the topology
generated by 
is equal to 
, that is, if 
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms
Kuratowski closure axioms
In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition...
.
Definition
A preclosure operator on a set is a map where
is the power set of .The preclosure operator has to satisfy the following properties:
-
(Preservation of nullary unions); -
(Extensivity); -
(Preservation of binary unions).
The last axiom implies the following:
- 4.
implies 
.
Topology
A set is closed (with respect to the preclosure) if . A set is open (with respect to the preclosure) if is closed. The collection of all open sets generated by the preclosure operator is a topologyTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
.
The closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
cl on this topological space satisfies
for all .Premetrics
Given a premetric on , thenis a preclosure on
.Sequential spaces
The sequential closure operator is a preclosure operator. Given a topology with respect to which the sequential closure operator is defined, the topological space is a sequential spaceSequential space
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability. Sequential spaces are the most general class of spaces for which sequences suffice to determine the topology....
if and only if the topology
generated by is equal to , that is, if .