This article is cited in 4 papers

The listing of topologies close to the discrete one on finite sets

I. G. Velichko^a, P. G. Stegantseva^b, N. P. Bashova<sup>b

a</sup> Chair of Higher Mathematics and Physics, Tavria State Agrotechnological University, 18 B. Khmelnitskogo Ave., Melitopol', Zaporozhye Region, 72310 Ukraine
^b Chair of Algebra and Geometry, Zaporozhye National University, 66 Zhukovskogo str., Zaporozhye, 69600 Ukraine

Abstract: We consider $T_0$-topologies on $n$-element set that contain more than $2^n$ elements. We solve a problem of listing and counting of such topologies. For this purpose we introduce the notions of an index of the topology and a vector of the topology. We study their properties and single out all possible classes of the topologies under the consideration. We formulate and prove a theorem related to the number of the topologies in each of the classes.

Keywords: topology on a finite set, $T_0$-topology, index of an element of a topology, index of a topology, vector of a topology.

UDC: 519.115

Received: 19.03.2014

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:11, 19–25

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