×

Subrings of a power set and clopen topologies. (English) Zbl 1486.13005

A clopen topology is a topology where every open set is also closed. In the first part of the paper, the authors study clopen topologies by showings that a topology is clopen if and only if its Kolmogoroff quotient is discrete, and provide a natural bijection between the clopen topologies on a set \(X\) and the set of all partitions of \(X\).
The second part of the paper connects topologies on \(X\) with the ring structure of the power set \(\mathcal{P}(X)\) of \(X\); in particular, every clopen topology is an example of a unitary subring of \(\mathcal{P}(X)\). The authors provide two constructions of classes of unitary subrings of \(\mathcal{P}(X)\), characterizing when these rings are topologies on \(X\).

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13F30 Valuation rings
54F65 Topological characterizations of particular spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexandroff, P., Diskrete Räume, Mat. S. (Novoya Seria)2 (1937) 501-518.
[2] Beer, G. and Bloomfield, C., A closure operator for clopen topologies, Bull. Belg. Math. Soc. Simon Stevin25 (2018) 149-159. · Zbl 1396.54003
[3] Grothendieck, A. and Dieudonné, J. A., Eléments de Géométrie Algébrique, , Vol. 166 (Springer-Verlag, Berlin, 1971). · Zbl 0203.23301
[4] Echi, O., Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)6 (2003) 489-507. · Zbl 1177.13060
[5] A. Facchini and C. A. Finocchiaro, Pretorsion theories, stable category and preordered sets, Annali di Matematica Pura ed Applicata (published online 2019), doi.org/10.1007/s10231-019-800912-2. · Zbl 1481.18002
[6] A. Facchini, C. A. Finocchiaro and M. Gran, Pretorsion theories in general categories, preprint (2019), arXiv:1908.03546. · Zbl 1457.18009
[7] A. Jaballah and N. Jarboui, From topologies of a set to subrings of its power set, Bull. Aust. Math. Soc. (published online 2020), doi:10.1017/S0004972720000015. · Zbl 1443.05187
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.