On the Kuratowski Closure-Complement Problem

Lilla Krystyna Baginska

University of Bialystok

Adam Grabowski

University of Bialystok

This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

Summary.

In this article we formalize the Kuratowski closure-complement result: there is at most 14 distinct sets that one can produce from a given subset $A$ of a topological space $T$ by applying closure and complement operators and that all 14 can be obtained from a suitable subset of $\Bbb R,$ namely KuratExSet $=\{1\} \cup {\Bbb Q} (2,3) \cup (3, 4)\cup (4,\infty)$.\par The second part of the article deals with the maximal number of distinct sets which may be obtained from a given subset $A$ of $T$ by applying closure and interior operators. The subset KuratExSet of $\Bbb R$ is also enough to show that 7 can be achieved.

MML Identifier: KURATO_1

Contents (PDF format)

1. Fourteen Kuratowski Sets
2. Seven Kuratowski Sets
3. The Set Generating Exactly Fourteen Kuratowski Sets
4. The Set Generating Exactly Seven Kuratowski Sets
5. The Difference Between Chosen Kuratowski Sets
6. Final Proofs For Seven Sets
7. Final Proofs For Fourteen Sets
8. Properties of Kuratowski Sets

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