Re: Comments on Norbert's topological extension of relational algebra

Op donderdag 17 december 2015 12:53:31 UTC+1 schreef Norbert_Paul:
> Jan Hidders wrote:
> >>>> True or false?
> >>>> Proposition. Let σ : 2^X → 2^X be a map (where 2^X is a powerset of X). Then, the following statements are equivalent.
> >>>> (1) σ is topological closure operator on X,
> >>>> (2) σ satisfies Kuratovski axiom, and there is a Galois connection (K, L) which satisfies σ = LK.
> >>>
> >>> False, I think. The identity would satisfy always (2) but not necessarily (1).
> >> I do not know much about Galois connections, but if identity can be constructed from a
> >> Galois conenction in the above manner then it is not a counter-example.
> >> Identity is the closure operator of the discrete topological space.
> >
> >
> > Ow, had not realised that. So on any discrete topology, the identity is a closure operator? And in fact the only closure operator?
> >
> > Jan Hidders
>
> Yes, each topological space has its unique closure operator.
> Different topologies give different closure operators.
>
> The discrete space: https://nl.wikipedia.org/wiki/Discrete_ruimte
> When you have an arbitrary set X, then the power set PX is a topology for X:
> Axiom 1: {} in PX, and X in PX
> Axiom 2: Every A, B in PX satisfies A \cap B \in PX.
> Axiom 3: For every subset S of PX the union \bigcap_{A \in S} S is an element of PX.
> The space (X, PX) is called the discrete space of X.
>
> The closure operator of the discrete space is
> cl(A) = A for all A subset X ,
> hence the identity function id: PX -> PX .
>
> Proof (easy):
> Let p be an arbitrary point in cl A. (Assumption)
> Then every open set Up that contains p as an
> element has a non-empty intersection:
> Up \cap A \neq {} (Definition of closure)
> Up = {p} is an element of PX (Definition of PX, Instanciation of Up)
> Hence {p} \cap A \neq {} (Consequence)
> Hence p \in {p} \cap A (YA Consequence)
> Hence p in A.
> Therefore cl A \subseteq A
> As A \subseteq cl A always holds we have:
>
> cl A = A . [qed]
>
> If the closure operator is not the identity function its topological space
> is not the discrete space.
>
> Other operators of the discrete space:
> Interior: int A = A for all A \subseteq X.
> Boundary: bd A = {} for all A \subseteq X.

Cool. Clear. Thank you for that explanation.

  Jan Hidders Received on Thu Dec 17 2015 - 17:46:29 CET