Topologies on the $n$-element set that consistent with close to the discrete topologies on $(n −1)$-element set

Abstract

UDC 519.1

Topologies on a finite set are described by a nondecreasing sequence of nonnegative integers (the vector of topologies). We study $T_0$ -topologies on the $n$-element set that induce topologies with $k > 2^{n - 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 $T_0$ -topologies with $k \geq 5 \cdot 2^{n - 4}$, which are described in works by Stanley and Kolli, and find the values $k \in [5 \cdot 2^{n - 4}, 2^{n - 1}]$, for which $T_0$ -topologies with k open sets do not exist. We describe all labeled $T_0$-topologies and indicate their number for each $k \geq 13 \cdot 2^{n - 5}$ . It is shown that there exist values $k \in (2^{n - 2}, 5 \cdot 2^{n - 4})$ such that any $T_0$ -topology with k open sets can not induce a topology close to the discrete one on an $(n - 1)$-element subset.