In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are de ...
, except that it is not required to be
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
. That is, a preclosure operator obeys only three of the four
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first forma ...
.
Definition
A preclosure operator on a set
is a map
:
where
is the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
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 topology; however, the above topology does not capture the notion of convergence associated to the operator, one should consider a
pretopology
In general topology, a pretopological space is a generalization of the concept of topological space.
A pretopological space can be defined in terms of either filters or a preclosure operator.
The similar, but more abstract, notion of a Grothendie ...
, instead.
[S. Dolecki, ''An Initiation into Convergence Theory'', in F. Mynard, E. Pearl (editors), ''Beyond Topology'',
AMS, Contemporary Mathematics, 2009.
]
Examples
Premetrics
Given
a
premetric on
, then
:
is 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 space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
if and only if the topology
generated by
is equal to
, that is, if
.
See also
*
Eduard Čech
Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...
References
* A.V. Arkhangelskii, L.S.Pontryagin, ''General Topology I'', (1990) Springer-Verlag, Berlin. {{ISBN, 3-540-18178-4.
* B. Banascheski
''Bourbaki's Fixpoint Lemma reconsidered'' Comment. Math. Univ. Carolinae 33 (1992), 303-309.
Closure operators