Topologies on the n-element set that consistent with close to the discrete topologies on (n1)-element set

Authors

DOI:

https://doi.org/10.37863/umzh.v73i2.6174

Abstract

UDC 519.1

Topologies on a finite set are described by a nondecreasing sequence of nonnegative integers (the vector of topologies). We study T0 -topologies on the n-element set that induce topologies with k>2n1 on the (n1)-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 T0 -topologies with k52n4, which are described in works by Stanley and Kolli, and find the values k[52n4,2n1], for which T0 -topologies with k open sets do not exist. We describe all labeled T0-topologies and indicate their number for each k132n5 . It is shown that there exist values k(2n2,52n4) such that any T0 -topology with k open sets can not induce a topology close to the discrete one on an (n1)-element subset.

References

J. W. Evans, F. Harary, M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10, № 5, 295 – 297 (1967). DOI: https://doi.org/10.1145/363282.363311

M. Kolli, Direct and elementary approach to enumerate topologies on a finite set, J. Integer Seq., 10, 1 – 11 (2007).

M. Kolli, On the cardinality of the T0 -topologies on a finite set, Int. J. Combin., 214, Article ID 798074 (2014), 7 p., https://doi.org/10.1155/2014/798074 DOI: https://doi.org/10.1155/2014/798074

V. Krishnamurthy, On the enumeration of homeomorphism classes of finite topologies, J. Austr. Math. Soc., Ser. A, 24, 320 – 338 (1977), https://doi.org/10.1017/s1446788700020358 DOI: https://doi.org/10.1017/S1446788700020358

Jr. H. Sharp, Quasi-orderings and topologies on finite sets, Proc. Amer. Math. Soc., 17, 1344 – 1349 (1966), https://doi.org/10.2307/2035738 DOI: https://doi.org/10.2307/2035738

Jr. H. Sharp, Cardinality of finite topologies, J. Combin. Theory, 5, 82 – 86 (1968). DOI: https://doi.org/10.1016/S0021-9800(68)80031-6

R.P. Stanley, On the number of open sets of finite topologies, J. Combin. Theory, 10, 74 – 79 (1971), https://doi.org/10.1016/0097-3165(71)90065-3 DOI: https://doi.org/10.1016/0097-3165(71)90065-3

D. Stephen, Topology on finite sets, Amer. Math. Monthly, 75, 739 – 741 (1968), https://doi.org/10.2307/2315186 DOI: https://doi.org/10.2307/2315186

N. P. Adamenko, I. G. Velichko, Klassifikacziya topologij na konechny`kh mnozhestvakh s pomoshh`yu grafov, Ukr. mat. zhurn., 60, № 7, 992 – 996 (2008).

Z. I. Borevich, K voprosu perechisleniya konechny`kh topologij, Zap. nauch. sem. LOMI, 71, 47 – 65 (1977).

I. G. Velichko, P. G. Steganczeva, N. P. Bashova, Perechislenie topologij blizkikh k diskretnoj na konechny`kh mnozhestvakh, Izv. vuzov. Matematika, № 11. 23 – 31 (2015). DOI: https://doi.org/10.1590/0102-4698136686

On-lajn e`ncziklopediya czelochislenny`kh posledovatel`nostej. URL: https://oeis.org/?language =russian.

Published

22.02.2021

Issue

Section

Research articles

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 (n1)-Element Set”. Ukrains’kyi Matematychnyi Zhurnal, vol. 73, no. 2, Feb. 2021, pp. 238-4, https://doi.org/10.37863/umzh.v73i2.6174.