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


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.


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How to Cite
Stegantseva, P. G., and A. V. Skryabina. “Topologies on the $n$-Element Set That Consistent With Close to the Discrete Topologies on $(n −1)$-Element Set”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 238 -48, doi:10.37863/umzh.v73i2.6174.
Research articles