Volume 15, 2003

University of Bialystok

Copyright (c) 2003 Association of Mizar Users

**Adam Grabowski**- University of Bialystok

- In the paper we give formal descriptions of the two Kuratowski limit oprators: Li $S$ and Ls $S$, where $S$ is an arbitrary sequence of subsets of a fixed topological space. In the two last sections we prove basic properties of these lower and upper topological limits, which may be found e.g. in [19]. In the sections 2-4, we present three operators which are associated in some sense with the above mentioned, that is lim inf $F$, lim sup $F$, and limes $F$, where $F$ is a sequence of subsets of a fixed 1-sorted structure.

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

- Preliminaries
- Lower and Upper Limit of Sequences of Subsets
- Ascending and Descending Families of Subsets
- Constant and Convergent Sequences
- Topological Lemmas
- Subsequences
- The Lower Topological Limit
- The Upper Topological Limit

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The ordinal numbers.
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Bartlomiej Skorulski.
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Miroslaw Wysocki and Agata Darmochwal.
Subsets of topological spaces.
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