preclosure operator

A preclosure operator on a set $X$ is a map $[\backslash \; \backslash \; ]\_p$

:$[\backslash \; \backslash \; ]\_p:\backslash mathcal\{P\}(X)\; \backslash to\; \backslash mathcal\{P\}(X)$

where $\backslash mathcal\{P\}(X)$ is the power set of $X.$

The preclosure operator has to satisfy the following properties:

1. $[\backslash varnothing]\_p\; =\; \backslash varnothing\; \backslash !$ (Preservation of nullary unions);
2. $A\; \backslash subseteq\; [A]\_p$ (Extensivity);
3. $[A\; \backslash cup\; B]\_p\; =\; [A]\_p\; \backslash cup\; [B]\_p$ (Preservation of binary unions).

The last axiom implies the following:

: 4. $A\; \backslash subseteq\; B$ implies $[A]\_p\; \backslash subseteq\; [B]\_p$.

A set $A$ is closed (with respect to the preclosure) if $[A]\_p=A$. A set $U\; \backslash subset\; X$ is open (with respect to the preclosure) if its complement $A\; =\; X\; \backslash setminus\; U$ is closed. The collection of all open sets generated by the preclosure operator is a topology;Eduard Čech, Zdeněk Frolík, Miroslav Katětov, Topological spaces Prague: Academia, Publishing House of the Czechoslovak Academy of

Sciences, 1966, Theorem 14 A.9 [https://eudml.org/doc/277000]. however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.S. Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology,

AMS, Contemporary Mathematics, 2009.

Given $d$ a premetric on $X$, then

:$[A]\_p\; =\; \backslash \{x\; \backslash in\; X\; :\; d(x,A)=0\backslash \}$

is a preclosure on $X.$

The sequential closure operator $[\backslash \; \backslash \; ]\_\backslash text\{seq\}$ is a preclosure operator. Given a topology $\backslash mathcal\{T\}$ with respect to which the sequential closure operator is defined, the topological space $(X,\backslash mathcal\{T\})$ is a sequential space if and only if the topology $\backslash mathcal\{T\}\_\backslash text\{seq\}$ generated by $[\backslash \; \backslash \; ]\_\backslash text\{seq\}$ is equal to $\backslash mathcal\{T\},$ that is, if $\backslash mathcal\{T\}\_\backslash text\{seq\}\; =\; \backslash mathcal\{T\}.$

• Eduard Čech

{{reflist}}

• A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. {{ISBN|3-540-18178-4}}.
• B. Banascheski, [http://www.emis.de/journals/CMUC/pdf/cmuc9202/banas.pdf Bourbaki's Fixpoint Lemma reconsidered], Comment. Math. Univ. Carolinae 33 (1992), 303–309.

Category:Closure operators