Aposyndesis in ℕ

José del Carmen Alberto-Domínguez; Gerardo Acosta; Maira Madriz-Mendoza

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 3, page 359-371
  • ISSN: 0010-2628

Abstract

top
We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions P ( a , b ) with the property that every prime number that divides a also divides b , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic in the Golomb space.

How to cite

top

Alberto-Domínguez, José del Carmen, Acosta, Gerardo, and Madriz-Mendoza, Maira. "Aposyndesis in $\mathbb {N}$." Commentationes Mathematicae Universitatis Carolinae 64.3 (2023): 359-371. <http://eudml.org/doc/299225>.

@article{Alberto2023,
abstract = {We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P(a,b)$ with the property that every prime number that divides $a$ also divides $b$, it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic in the Golomb space.},
author = {Alberto-Domínguez, José del Carmen, Acosta, Gerardo, Madriz-Mendoza, Maira},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {aposyndesis; arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space},
language = {eng},
number = {3},
pages = {359-371},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Aposyndesis in $\mathbb \{N\}$},
url = {http://eudml.org/doc/299225},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Alberto-Domínguez, José del Carmen
AU - Acosta, Gerardo
AU - Madriz-Mendoza, Maira
TI - Aposyndesis in $\mathbb {N}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 3
SP - 359
EP - 371
AB - We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P(a,b)$ with the property that every prime number that divides $a$ also divides $b$, it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are aposyndetic in the Golomb space.
LA - eng
KW - aposyndesis; arithmetic progression; Golomb topology; Kirch topology; totally Brown space; totally separated space
UR - http://eudml.org/doc/299225
ER -

References

top
  1. Alberto-Domínguez J. D. C., Acosta G., Delgadillo-Piñón G., Totally Brown subsets of the Golomb space and the Kirch space, Comment. Math. Univ. Carolin. 63 (2022), no. 2, 189–219. MR4506132
  2. Alberto-Domínguez J. D. C., Acosta G., Madriz-Mendoza M., The common division topology on ℕ , Comment. Math. Univ. Carolin. 63 (2022), no. 3, 329–349. MR4542793
  3. 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. MR3914710
  4. Engelking R., General Topology, Sigma Ser. Pure Math., 6, Heldermann Verlag, Berlin, 1989. Zbl0684.54001MR1039321
  5. Golomb S. W., 10.1080/00029890.1959.11989385, Amer. Math. Monthly 66 (1959), 663–665. MR0107622DOI10.1080/00029890.1959.11989385
  6. Golomb S. W., Arithmetica topologica, General Topology and Its Relations to Modern Analysis and Algebra, Proc. Sympos., Prague, 1961, Academic Press, New York, 1961, pages 179–186 (Italian). MR0154249
  7. Kirch A. M., 10.1080/00029890.1969.12000163, Amer. Math. Monthly 76 (1969), 169–171. MR0239563DOI10.1080/00029890.1969.12000163
  8. Steen L. A., Seebach J. A., Jr., Counterexamples in Topology, Dover Publications, Mineola, New York, 1995. Zbl0386.54001MR1382863
  9. Szczuka P., 10.1515/dema-2010-0416, Demonstratio Math. 43 (2010), no. 4, 899–909. MR2761648DOI10.1515/dema-2010-0416

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.