On the Kuratowski Limit Operators
Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003
Association of Mizar Users
On the Kuratowski Limit Operators
-
Adam Grabowski
-
University of Bialystok
Summary.
-
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.
The terminology and notation used in this paper have been
introduced in the following articles
[29]
[33]
[2]
[32]
[9]
[1]
[22]
[24]
[35]
[12]
[34]
[6]
[4]
[18]
[8]
[7]
[16]
[5]
[13]
[25]
[30]
[21]
[10]
[23]
[14]
[15]
[20]
[17]
[27]
[28]
[26]
[11]
[3]
[31]
-
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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Received August 12, 2003
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