| Title:
|
Totally Brown subsets of the Golomb space and the Kirch space (English) |
| Author:
|
Alberto-Domínguez, José del Carmen |
| Author:
|
Acosta, Gerardo |
| Author:
|
Delgadillo-Piñón, Gerardo |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2022 |
| Pages:
|
189-219 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A topological space $X$ is totally Brown if for each $n \in \mathbb{N} \setminus \{1\}$ and every nonempty open subsets $U_1,U_2,\ldots,U_n$ of $X$ we have ${\rm cl}_X(U_1) \cap {\rm cl}_X(U_2) \cap \cdots \cap {\rm cl}_X(U_n) \ne \emptyset$. Totally Brown spaces are connected. In this paper we consider the Golomb topology $\tau_G$ on the set $\mathbb{N}$ of natural numbers, as well as the Kirch topology $\tau_K$ on $\mathbb{N}$. Then we examine subsets of these spaces which are totally Brown. Among other results, we characterize the arithmetic progressions which are either totally Brown or totally separated in $(\mathbb{N},\tau_G)$. We also show that $(\mathbb{N},\tau_G)$ and $(\mathbb{N},\tau_K)$ are aposyndetic. Our results generalize properties obtained by A. M. Kirch in 1969 and by P. Szczuka in 2010, 2013 and 2015. (English) |
| Keyword:
|
arithmetic progression |
| Keyword:
|
Golomb topology |
| Keyword:
|
Kirch topology |
| Keyword:
|
totally Brown space |
| Keyword:
|
totally separated space |
| MSC:
|
11A41 |
| MSC:
|
11B05 |
| MSC:
|
11B25 |
| MSC:
|
54A05 |
| MSC:
|
54D05 |
| MSC:
|
54D10 |
| idZBL:
|
Zbl 07613030 |
| idMR:
|
MR4506132 |
| DOI:
|
10.14712/1213-7243.2022.017 |
| . |
| Date available:
|
2022-11-02T09:16:56Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151085 |
| . |
| Reference:
|
[1] Alberto-Domínguez J. C., Acosta G., Madriz-Mendoza M.: The common division topology on $\mathbb{N}$.accepted in Comment. Math. Univ. Carolin. |
| Reference:
|
[2] Banakh T., Mioduszewski J., Turek S.: On continuous self-maps and homeomorphisms of the Golomb space.Comment. Math. Univ. Carolin. 59 (2018), no. 4, 423–442. |
| Reference:
|
[3] Banakh T., Stelmakh Y., Turek S.: The Kirch space is topologically rigid.Topology Appl. 304 (2021), Paper No. 107782, 16 pages. MR 4339625, 10.1016/j.topol.2021.107782 |
| Reference:
|
[4] Bing R. H.: A connected countable Hausdorff space.Proc. Amer. Math. Soc. 4 (1953), 474. MR 0060806, 10.1090/S0002-9939-1953-0060806-9 |
| Reference:
|
[5] Clark P. L., Lebowitz-Lockard N., Pollack P.: A note on Golomb topologies.Quaest. Math. 42 (2019), no. 1, 73–86. MR 3905659, 10.2989/16073606.2018.1438533 |
| Reference:
|
[6] Dontchev J.: On superconnected spaces.Serdica 20 (1994), no. 3–4, 345–350. MR 1333356 |
| Reference:
|
[7] Engelking R.: General Topology.Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[8] Fine B., Rosenberger G.: Number Theory. An Introduction via the Density of Primes.Birkhäuser/Springer, Cham, 2016. MR 3559913 |
| Reference:
|
[9] Furstenberg H.: On the infinitude of primes.Amer. Math. Monthly 62 (1955), 353. MR 0068566, 10.2307/2307043 |
| Reference:
|
[10] Golomb S. W.: A connected topology for the integers.Amer. Math. Monthly 66 (1959), 663–665. MR 0107622, 10.1080/00029890.1959.11989385 |
| Reference:
|
[11] Golomb S. W.: Arithmetica topologica.General Topology and Its Relations to Modern Analysis and Algebra, Proc. Sympos., Prague, 1961, Academic Press, New York, Publ. House Czech. Acad. Sci., Praha, 1962, 179–186 (Italian). MR 0154249 |
| Reference:
|
[12] Jones F. B.: Aposyndetic continua and certain boundary problems.Amer. J. Math. 63 (1941), 545–553. MR 0004771, 10.2307/2371367 |
| Reference:
|
[13] Jones G. A., Jones J. M.: Elementary Number Theory.Springer Undergraduate Mathematics Series, Springer, London, 1998. MR 1610533 |
| Reference:
|
[14] Kirch A. M.: A countable, connected, locally connected Hausdorff space.Amer. Math. Monthly 76 (1969), 169–171. MR 0239563, 10.1080/00029890.1969.12000163 |
| Reference:
|
[15] Nanda S., Panda H. K.: The fundamental group of principal superconnected spaces.Rend. Mat. (6) 9 (1976), no. 4, 657–664. MR 0434295 |
| Reference:
|
[16] Rizza G. B.: A topology for the set of nonnegative integers.Riv. Mat. Univ. Parma (5) 2 1993, 179–185. MR 1276050 |
| Reference:
|
[17] Steen L. A., Seebach J. A., Jr.: Counterexamples in Topology.Dover Publications, Mineola, New York, 1995. Zbl 0386.54001, MR 1382863 |
| Reference:
|
[18] Szczuka P.: The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies.Demonstratio Math. 43 (2010), no. 4, 899–909. MR 2761648, 10.1515/dema-2010-0416 |
| Reference:
|
[19] Szczuka P.: Connections between connected topological spaces on the set of positive integers.Cent. Eur. J. Math. 11 (2013), no. 5, 876–881. MR 3032336 |
| Reference:
|
[20] Szczuka P.: The Darboux property for polynomials in Golomb's and Kirch's topologies.Demonstratio Math. 46 (2013), no. 2, 429–435. MR 3098036 |
| Reference:
|
[21] Szczuka P.: Regular open arithmetic progressions in connected topological spaces on the set of positive integers.Glas. Mat. Ser. III 49(69) (2014), no. 1, 13–23. MR 3224474, 10.3336/gm.49.1.02 |
| Reference:
|
[22] Szczuka P.: The closures of arithmetic progressions in the common division topology on the set of positive integers.Cent. Eur. J. Math. 12 (2014), no. 7, 1008–1014. MR 3188461 |
| Reference:
|
[23] Szczuka P.: The closures of arithmetic progressions in Kirch's topology on the set of positive integers.Int. J. Number Theory 11 (2015), no. 3, 673–682. MR 3327837, 10.1142/S1793042115500360 |
| Reference:
|
[24] Szyszkowska P., Szyszkowski M.: Properties of the common division topology on the set of positive integers.J. Ramanujan Math. Soc. 33 (2018), no. 1, 91–98. MR 3772612 |
| . |
