Geodesic
The listing of topologies close to the discrete one on finite sets
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 23-31

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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. 
@article{IVM_2015_11_a1,
     author = {I. G. Velichko and P. G. Stegantseva and N. P. Bashova},
     title = {The listing of topologies close to the discrete one on finite sets},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {23--31},
     publisher = {mathdoc},
     number = {11},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2015_11_a1/}
}
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I. G. Velichko; P. G. Stegantseva; N. P. Bashova. The listing of topologies close to the discrete one on finite sets. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 23-31. http://geodesic.mathdoc.fr/item/IVM_2015_11_a1/