Spaces not distinguishing pointwise and ℐ -quasinormal convergence

Pratulananda Das; Debraj Chandra

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 1, page 83-96
  • ISSN: 0010-2628

Abstract

top
In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of ℐ -quasinormal convergence. We then introduce the notion of ℐ Q N ( ℐ w Q N ) space as a topological space in which every sequence of continuous real valued functions pointwise converging to 0 , is also ℐ -quasinormally convergent to 0 (has a subsequence which is ℐ -quasinormally convergent to 0 ) and make certain observations on those spaces.

How to cite

top

Das, Pratulananda, and Chandra, Debraj. "Spaces not distinguishing pointwise and $\mathcal {I}$-quasinormal convergence." Commentationes Mathematicae Universitatis Carolinae 54.1 (2013): 83-96. <http://eudml.org/doc/252493>.

@article{Das2013,
abstract = {In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $\mathcal \{I\}$-quasinormal convergence. We then introduce the notion of $\mathcal \{I\}QN (\mathcal \{I\}wQN)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $\mathcal \{I\}$-quasinormally convergent to $0$ (has a subsequence which is $\mathcal \{I\}$-quasinormally convergent to $0$) and make certain observations on those spaces.},
author = {Das, Pratulananda, Chandra, Debraj},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {ideal; filter; $\mathcal \{I\}$-quasinormal convergence; Chain Condition; $AP$-ideal; $\mathcal \{I\}QN$ space; $\mathcal \{I\}wQN$ space; -quasinormal convergence; countably generated ideal; -ideal; space; space},
language = {eng},
number = {1},
pages = {83-96},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Spaces not distinguishing pointwise and $\mathcal \{I\}$-quasinormal convergence},
url = {http://eudml.org/doc/252493},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Das, Pratulananda
AU - Chandra, Debraj
TI - Spaces not distinguishing pointwise and $\mathcal {I}$-quasinormal convergence
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 1
SP - 83
EP - 96
AB - In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $\mathcal {I}$-quasinormal convergence. We then introduce the notion of $\mathcal {I}QN (\mathcal {I}wQN)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $\mathcal {I}$-quasinormally convergent to $0$ (has a subsequence which is $\mathcal {I}$-quasinormally convergent to $0$) and make certain observations on those spaces.
LA - eng
KW - ideal; filter; $\mathcal {I}$-quasinormal convergence; Chain Condition; $AP$-ideal; $\mathcal {I}QN$ space; $\mathcal {I}wQN$ space; -quasinormal convergence; countably generated ideal; -ideal; space; space
UR - http://eudml.org/doc/252493
ER -

References

top
  1. Balcar B., Pelant J., Simon P., The space of ultrafilters on ℕ covered by nowhere dense sets, Fund. Math. 110 (1980), 11–24. MR0600576
  2. Balcerzak M., Dems K., Komisarski A., 10.1016/j.jmaa.2006.05.040, J. Math. Anal. Appl. 328 (2007), no. 1, 715–729. MR2285579DOI10.1016/j.jmaa.2006.05.040
  3. Bukovská Z., Thin sets in trigonometrical series and quasinormal convergence, Math. Slovaca 40 (1990), 53–62. Zbl0733.43003MR1094972
  4. Bukovská Z., Quasinormal convergence, Math. Slovaca 41 (1991), no. 2, 137–146. Zbl0757.40004MR1108577
  5. Bukovský L., Reclaw I., Repický M., 10.1016/0166-8641(91)90098-7, Topology Appl. 41 (1991), no. 1–2, 25–40. Zbl0768.54025MR1129696DOI10.1016/0166-8641(91)90098-7
  6. Bukovský L., Reclaw I., Repický M., 10.1016/S0166-8641(99)00226-6, Topology Appl. 112 (2001), no. 1, 13–40. MR1815270DOI10.1016/S0166-8641(99)00226-6
  7. Bukovský L., Haleš J., 10.1016/j.topol.2006.09.008, Topology Appl. 154 (2007), no. 4, 848–858. Zbl1117.54003MR2294632DOI10.1016/j.topol.2006.09.008
  8. Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463–472. Zbl0405.26006MR0515347
  9. Császár Á., Laczkovich M., 10.1007/BF01903381, Acta. Math. Acad. Sci. Hungar. 33 (1979), 51–70. Zbl0401.54010MR0515120DOI10.1007/BF01903381
  10. Chandra D., Das P., Some further investigations of open covers and selection principles using ideals, Topology Proc. 39 (2012), 281–291. MR2869444
  11. Das P., Ghosal S.K., 10.1016/j.camwa.2010.01.027, Comput. Math. Appl. 59 (2010), no. 8, 2597–2600. MR2607963DOI10.1016/j.camwa.2010.01.027
  12. Das P., Ghosal S.K., 10.1016/j.topol.2010.02.003, Topology Appl. 157 (2010), no. 7, 1152–1156. MR2607080DOI10.1016/j.topol.2010.02.003
  13. Das P., Ghosal S.K., 10.1016/j.topol.2011.05.006, Topology Appl. 158 (2011), no. 13, 1529–1533. MR2812462DOI10.1016/j.topol.2011.05.006
  14. Das P., 10.1016/j.topol.2012.04.007, Topology Appl. 159 (2012), 2621–2625. Zbl1250.40002MR2923430DOI10.1016/j.topol.2012.04.007
  15. Das P., Dutta S., On some types of convergence of sequences of functions in ideal context, Filomat 27 (2013), no. 1, 147–154. 
  16. Das P., Certain types of covers and selection principles using ideals, Houston J. Math. 39 (2013), no. 2, 447–460. 
  17. Denjoy A., Leçons sur le calcul des coefficients d’une série trigonométrique, 2 e partie, Paris, 1941. 
  18. Di Maio G., Kočinac Lj.D.R., 10.1016/j.topol.2008.01.015, Topology Appl. 156 (2008), 28–45. Zbl1155.54004MR2463821DOI10.1016/j.topol.2008.01.015
  19. Fast H., Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. Zbl0044.33605MR0048548
  20. Fridy J.A., On statistical convergence, Analysis 5 (1985), 301–313. Zbl0588.40001MR0816582
  21. Gerlits J., Nagy Z., Some properties of C ( X ) , I, Topology Appl. 14 (1984), 145–155. Zbl0503.54020MR0667661
  22. Jacobs K., Measure and Integral, Academic Press, New York-London, 1978. Zbl0446.28001MR0514702
  23. Kostyrko P., Šalát T., Wilczyński W., ℐ -convergence, Real Anal. Exchange 26 (2000/2001), no. 2, 669–685. MR1844385
  24. Komisarski A., 10.1016/j.jmaa.2007.09.016, J. Math. Anal. Appl. 340 (2008), 770–779. MR2390885DOI10.1016/j.jmaa.2007.09.016
  25. Lahiri B.K., Das P., ℐ and ℐ * -convergence in topological spaces, Math. Bohemica 130 (2005), 153–160. MR2148648
  26. Lahiri B.K., Das P., ℐ and ℐ * -convergence of nets, Real Anal. Exchange 33 (2008), no. 2, 431–442. MR2458259
  27. Mro.zek N., 10.1016/j.jmaa.2008.08.032, J. Math. Anal. Appl. 349 (2009), 452–458. MR2456202DOI10.1016/j.jmaa.2008.08.032
  28. Papanastassiou N., On a new type of convergence of sequences of functions, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), no. 2, 493–506. Zbl1221.28012MR1958294
  29. Šalát T., On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150. Zbl0437.40003MR0587239
  30. Schoenberg I.J., 10.2307/2308747, Amer. Math. Monthly 66 (1959), 361–375. Zbl0089.04002MR0104946DOI10.2307/2308747
  31. Van Douven E.K., The integers and topology, Handbook of Set-theoritic Topology, K. Kunen and J.E. Vaughan (eds.), North-Holland, Amsterdam, 1984. MR0776622

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.