Totally Brown subsets of the Golomb space and the Kirch space

José del Carmen Alberto-Domínguez; Gerardo Acosta; Gerardo Delgadillo-Piñón

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 2, page 189-219
  • ISSN: 0010-2628

Abstract

top
A topological space X is totally Brown if for each n ∈ ℕ ∖ { 1 } and every nonempty open subsets U 1 , U 2 , ... , U n of X we have cl X ( U 1 ) ∩ cl X ( U 2 ) ∩ ⋯ ∩ cl X ( U n ) ≠ ∅ . Totally Brown spaces are connected. In this paper we consider the Golomb topology τ G on the set ℕ of natural numbers, as well as the Kirch topology τ K on 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 ( ℕ , τ G ) . We also show that ( ℕ , τ G ) and ( ℕ , τ K ) are aposyndetic. Our results generalize properties obtained by A. M. Kirch in 1969 and by P. Szczuka in 2010, 2013 and 2015.

How to cite

top

Alberto-Domínguez, José del Carmen, Acosta, Gerardo, and Delgadillo-Piñón, Gerardo. "Totally Brown subsets of the Golomb space and the Kirch space." Commentationes Mathematicae Universitatis Carolinae 62 63.2 (2022): 189-219. <http://eudml.org/doc/298517>.

@article{Alberto2022,
abstract = {A topological space $X$ is totally Brown if for each $n \in \mathbb \{N\} \setminus \lbrace 1\rbrace $ 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.},
author = {Alberto-Domínguez, José del Carmen, Acosta, Gerardo, Delgadillo-Piñón, Gerardo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space},
language = {eng},
number = {2},
pages = {189-219},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Totally Brown subsets of the Golomb space and the Kirch space},
url = {http://eudml.org/doc/298517},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Alberto-Domínguez, José del Carmen
AU - Acosta, Gerardo
AU - Delgadillo-Piñón, Gerardo
TI - Totally Brown subsets of the Golomb space and the Kirch space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 2
SP - 189
EP - 219
AB - A topological space $X$ is totally Brown if for each $n \in \mathbb {N} \setminus \lbrace 1\rbrace $ 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.
LA - eng
KW - arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
UR - http://eudml.org/doc/298517
ER -

References

top
  1. Alberto-Domínguez J. C., Acosta G., Madriz-Mendoza M., The common division topology on ℕ , accepted in Comment. Math. Univ. Carolin. 
  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. 
  3. Banakh T., Stelmakh Y., Turek S., 10.1016/j.topol.2021.107782, Topology Appl. 304 (2021), Paper No. 107782, 16 pages. MR4339625DOI10.1016/j.topol.2021.107782
  4. Bing R. H., 10.1090/S0002-9939-1953-0060806-9, Proc. Amer. Math. Soc. 4 (1953), 474. MR0060806DOI10.1090/S0002-9939-1953-0060806-9
  5. Clark P. L., Lebowitz-Lockard N., Pollack P., 10.2989/16073606.2018.1438533, Quaest. Math. 42 (2019), no. 1, 73–86. MR3905659DOI10.2989/16073606.2018.1438533
  6. Dontchev J., On superconnected spaces, Serdica 20 (1994), no. 3–4, 345–350. MR1333356
  7. Engelking R., General Topology, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Zbl0684.54001MR1039321
  8. Fine B., Rosenberger G., Number Theory. An Introduction via the Density of Primes, Birkhäuser/Springer, Cham, 2016. MR3559913
  9. Furstenberg H., 10.2307/2307043, Amer. Math. Monthly 62 (1955), 353. MR0068566DOI10.2307/2307043
  10. Golomb S. W., 10.1080/00029890.1959.11989385, Amer. Math. Monthly 66 (1959), 663–665. MR0107622DOI10.1080/00029890.1959.11989385
  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). MR0154249
  12. Jones F. B., 10.2307/2371367, Amer. J. Math. 63 (1941), 545–553. MR0004771DOI10.2307/2371367
  13. Jones G. A., Jones J. M., Elementary Number Theory, Springer Undergraduate Mathematics Series, Springer, London, 1998. MR1610533
  14. Kirch A. M., 10.1080/00029890.1969.12000163, Amer. Math. Monthly 76 (1969), 169–171. MR0239563DOI10.1080/00029890.1969.12000163
  15. Nanda S., Panda H. K., The fundamental group of principal superconnected spaces, Rend. Mat. (6) 9 (1976), no. 4, 657–664. MR0434295
  16. Rizza G. B., A topology for the set of nonnegative integers, Riv. Mat. Univ. Parma (5) 2 1993, 179–185. MR1276050
  17. Steen L. A., Seebach J. A., Jr., Counterexamples in Topology, Dover Publications, Mineola, New York, 1995. Zbl0386.54001MR1382863
  18. Szczuka P., 10.1515/dema-2010-0416, Demonstratio Math. 43 (2010), no. 4, 899–909. MR2761648DOI10.1515/dema-2010-0416
  19. Szczuka P., Connections between connected topological spaces on the set of positive integers, Cent. Eur. J. Math. 11 (2013), no. 5, 876–881. MR3032336
  20. Szczuka P., The Darboux property for polynomials in Golomb's and Kirch's topologies, Demonstratio Math. 46 (2013), no. 2, 429–435. MR3098036
  21. Szczuka P., 10.3336/gm.49.1.02, Glas. Mat. Ser. III 49(69) (2014), no. 1, 13–23. MR3224474DOI10.3336/gm.49.1.02
  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. MR3188461
  23. Szczuka P., 10.1142/S1793042115500360, Int. J. Number Theory 11 (2015), no. 3, 673–682. MR3327837DOI10.1142/S1793042115500360
  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. MR3772612

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.