Topologies on the n-element set that consistent with close to the discrete topologies on (n−1)-element set
DOI:
https://doi.org/10.37863/umzh.v73i2.6174Abstract
UDC 519.1
Topologies on a finite set are described by a nondecreasing sequence of nonnegative integers (the vector of topologies). We study T0 -topologies on the n-element set that induce topologies with k>2n−1 on the (n−1)-element set (these induced topologies are called close to the discrete topology). Let k denote the number of open sets in a topology. We obtain the form of the vector of T0 -topologies with k≥5⋅2n−4, which are described in works by Stanley and Kolli, and find the values k∈[5⋅2n−4,2n−1], for which T0 -topologies with k open sets do not exist. We describe all labeled T0-topologies and indicate their number for each k≥13⋅2n−5 . It is shown that there exist values k∈(2n−2,5⋅2n−4) such that any T0 -topology with k open sets can not induce a topology close to the discrete one on an (n−1)-element subset.
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