N. Elez and O. Papaz, [The new operators in topological spaces,
Math. Moravica 17 (2013), 63--68] stated this fact without any proof.
In the attached pdf, I have been able to worked out a
difficult proof for a straightforward generalization of this fact.
Could somebody suggest me some simplifications in my proof, or
some nicely written, modern literature on this topic.
As I remember, only the heavy books of Kuratowski and Cech are
using closure as a strarting point instead of open sets.
Though, closures are more powerful tools than open sets.
In my opinion, open sets are the greatest mistake of topologists.
Curiously enough, the operations defined by
res(A)=cl(A)--A and bnd(A)=cl(A)--int(A) are already not idempotent
even if cl and int are so.
Dear Colleagues,
Meantime, by using a simple, but useful set-theoretic trick,
we could simplify the proof of Theorem 3.3 of Extract-1.pdf.
For some instructive observations, see the attached Observ.pdf
which have not been checked by my coauthors.
Further simplifications, illustrating examples, applications,
generalizations, and critical discussions are still wellcomed.
Some important generalizations can be obtained by using
proximal closures and interiors instead of the topological ones.
With best regards,
Arpad Szaz
- Badges
- Science topic
- Similar topics
- Philosophy
- Philosophy Of Science
- Reasoning