Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Lilla Krystyna Baginska**- University of Bialystok
**Adam Grabowski**- University of Bialystok
- This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

- 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.

- Fourteen Kuratowski Sets
- Seven Kuratowski Sets
- The Set Generating Exactly Fourteen Kuratowski Sets
- The Set Generating Exactly Seven Kuratowski Sets
- The Difference Between Chosen Kuratowski Sets
- Final Proofs For Seven Sets
- Final Proofs For Fourteen Sets
- Properties of Kuratowski Sets

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