Topologies on the n-Element Set Consistent with Topologies Close to Discrete on an (n−1)-Element Set

Topologies on a finite set are described by a nondecreasing sequence of nonnegative integers (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). By k we denote the number of open sets in a topology. We obtain the form of the vector of T₀-topologies with k ≥ 5 · 2ⁿ⁻⁴ described by Stanley and Kolli and find the values k 𝜖 [5 · 2ⁿ⁻⁴, 2ⁿ⁻¹] for which the 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 𝜖 (2ⁿ⁻², 5 · 2ⁿ⁻⁴) such that any T₀-topology with k open sets cannot induce a topology close to the discrete topology on an (n − 1)-element subset.