Praclosure operator
From Wikipedia, the free encyclopedia
In topology, a praclosure operator, or Čech closure operator is a map between subsets of a set, similar to a closure operator, except that it is not required to be idempotent. That is, a praclosure operator obeys only three of the four Kuratowski closure axioms.
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[edit] Definition
A praclosure operator on a set X is a map ![[\quad]_p](../../../math/9/e/6/9e6d26d0199acb41ad1ce8679db50349.png)
![[\quad]_p:\mathcal{P}(X) \to \mathcal{P}(X)](../../../math/0/4/8/04857e1eb8155fad436bcf40b6bf5a26.png)
where 
is the power set of X.
The praclosure operator has to satisfy the following properties:
![[\varnothing]_p = \varnothing \!](../../../math/2/c/9/2c940a4543457aa4a839f69a5db13005.png)
(Preservation of nullary unions)![A \subseteq [A]_p](../../../math/6/d/3/6d31c086c8911c09c12caad3b3f41e1a.png)
(Extensivity)![[A \cup B]_p = [A]_p \cup [B]_p](../../../math/a/c/e/ace6b170ed061af0a6126b104d4ff549.png)
(Preservation of binary unions)
[edit] Topology
A set A is closed (with respect to the praclosure) if [A]p = A. A set 
is open (with respect to the praclosure) if 
is closed. The collection of all open sets generated by the praclosure operator is a topology.
The closure operator cl on this topological space satisfies ![[A]_p\subseteq \operatorname{cl}(A)](../../../math/6/0/a/60ac18e93a990eca5d602c5b0a077c9a.png)
for all 
.
[edit] Examples
[edit] Prametrics
Given d a prametric on X, then
![[A]_p=\{x\in X : d(x,A)=0\}](../../../math/7/c/1/7c1fd6c332cbf5fb396fa1e06833a0d4.png)
is a praclosure on X.
[edit] Sequential spaces
The sequential closure operator ![[\quad]_\mbox{seq}](../../../math/3/f/1/3f196661654e75ba7503a476b26effea.png)
is a praclosure 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 
.
[edit] References
- A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4
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