| Title:
|
Spaces not distinguishing pointwise and $\mathcal{I}$-quasinormal convergence (English) |
| Author:
|
Das, Pratulananda |
| Author:
|
Chandra, Debraj |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
83-96 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
ideal |
| Keyword:
|
filter |
| Keyword:
|
$\mathcal{I}$-quasinormal convergence |
| Keyword:
|
Chain Condition |
| Keyword:
|
$AP$-ideal |
| Keyword:
|
$\mathcal{I}QN$ space |
| Keyword:
|
$\mathcal{I}wQN$ space |
| MSC:
|
40G15 |
| MSC:
|
54C30 |
| MSC:
|
54G99 |
| idMR:
|
MR3038073 |
| . |
| Date available:
|
2013-02-21T14:06:41Z |
| Last updated:
|
2015-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143154 |
| . |
| Reference:
|
[1] Balcar B., Pelant J., Simon P.: The space of ultrafilters on $\mathbb{N}$ covered by nowhere dense sets.Fund. Math. 110 (1980), 11–24. MR 0600576 |
| Reference:
|
[2] Balcerzak M., Dems K., Komisarski A.: Statistical convergence and ideal convergence for sequence of functions.J. Math. Anal. Appl. 328 (2007), no. 1, 715–729. MR 2285579, 10.1016/j.jmaa.2006.05.040 |
| Reference:
|
[3] Bukovská Z.: Thin sets in trigonometrical series and quasinormal convergence.Math. Slovaca 40 (1990), 53–62. Zbl 0733.43003, MR 1094972 |
| Reference:
|
[4] Bukovská Z.: Quasinormal convergence.Math. Slovaca 41 (1991), no. 2, 137–146. Zbl 0757.40004, MR 1108577 |
| Reference:
|
[5] Bukovský L., Reclaw I., Repický M.: Spaces not distinguishing pointwise and quasinormal convergence of real functions.Topology Appl. 41 (1991), no. 1–2, 25–40. Zbl 0768.54025, MR 1129696, 10.1016/0166-8641(91)90098-7 |
| Reference:
|
[6] Bukovský L., Reclaw I., Repický M.: Spaces not distinguishing convergence of real valued functions.Topology Appl. 112 (2001), no. 1, 13–40. MR 1815270, 10.1016/S0166-8641(99)00226-6 |
| Reference:
|
[7] Bukovský L., Haleš J.: $QN$ spaces, $wQN$ spaces and covering properties.Topology Appl. 154 (2007), no. 4, 848–858. Zbl 1117.54003, MR 2294632, 10.1016/j.topol.2006.09.008 |
| Reference:
|
[8] Császár Á., Laczkovich M.: Discrete and equal convergence.Studia Sci. Math. Hungar. 10 (1975), 463–472. Zbl 0405.26006, MR 0515347 |
| Reference:
|
[9] Császár Á., Laczkovich M.: Some remarks on discrete Baire classes.Acta. Math. Acad. Sci. Hungar. 33 (1979), 51–70. Zbl 0401.54010, MR 0515120, 10.1007/BF01903381 |
| Reference:
|
[10] Chandra D., Das P.: Some further investigations of open covers and selection principles using ideals.Topology Proc. 39 (2012), 281–291. MR 2869444 |
| Reference:
|
[11] Das P., Ghosal S.K.: Some further results on $\mathcal{I}$-Cauchy sequences and condition (AP).Comput. Math. Appl. 59 (2010), no. 8, 2597–2600. MR 2607963, 10.1016/j.camwa.2010.01.027 |
| Reference:
|
[12] Das P., Ghosal S.K.: On $\mathcal{I}$-Cauchy nets and completeness.Topology Appl. 157 (2010), no. 7, 1152–1156. MR 2607080, 10.1016/j.topol.2010.02.003 |
| Reference:
|
[13] Das P., Ghosal S.K.: When $\mathcal{I}$-Cauchy nets in complete uniform spaces are $\mathcal{I}$-convergent.Topology Appl. 158 (2011), no. 13, 1529–1533. MR 2812462, 10.1016/j.topol.2011.05.006 |
| Reference:
|
[14] Das P.: Some further results on ideal convergence in topological spaces.Topology Appl. 159 (2012), 2621–2625. Zbl 1250.40002, MR 2923430, 10.1016/j.topol.2012.04.007 |
| Reference:
|
[15] Das P., Dutta S.: On some types of convergence of sequences of functions in ideal context.Filomat 27 (2013), no. 1, 147–154. |
| Reference:
|
[16] Das P.: Certain types of covers and selection principles using ideals.Houston J. Math. 39 (2013), no. 2, 447–460. |
| Reference:
|
[17] Denjoy A.: Leçons sur le calcul des coefficients d'une série trigonométrique, $2^e$ partie.Paris, 1941. |
| Reference:
|
[18] Di Maio G., Kočinac Lj.D.R.: Statistical convergence in topology.Topology Appl. 156 (2008), 28–45. Zbl 1155.54004, MR 2463821, 10.1016/j.topol.2008.01.015 |
| Reference:
|
[19] Fast H.: Sur la convergence statistique.Colloq. Math. 2 (1951), 241–244. Zbl 0044.33605, MR 0048548 |
| Reference:
|
[20] Fridy J.A.: On statistical convergence.Analysis 5 (1985), 301–313. Zbl 0588.40001, MR 0816582 |
| Reference:
|
[21] Gerlits J., Nagy Z.: Some properties of $C(X)$, I.Topology Appl. 14 (1984), 145–155. Zbl 0503.54020, MR 0667661 |
| Reference:
|
[22] Jacobs K.: Measure and Integral.Academic Press, New York-London, 1978. Zbl 0446.28001, MR 0514702 |
| Reference:
|
[23] Kostyrko P., Šalát T., Wilczyński W.: $\mathcal{I}$-convergence.Real Anal. Exchange 26 (2000/2001), no. 2, 669–685. MR 1844385 |
| Reference:
|
[24] Komisarski A.: Pointwise $\mathcal{I}$-convergence and $\mathcal{I}$-convergence in measure of sequences of functions.J. Math. Anal. Appl. 340 (2008), 770–779. MR 2390885, 10.1016/j.jmaa.2007.09.016 |
| Reference:
|
[25] Lahiri B.K., Das P.: $\mathcal{I}$ and $\mathcal{I}^*$-convergence in topological spaces.Math. Bohemica 130 (2005), 153–160. MR 2148648 |
| Reference:
|
[26] Lahiri B.K., Das P.: $\mathcal{I}$ and $\mathcal{I}^*$-convergence of nets.Real Anal. Exchange 33 (2008), no. 2, 431–442. MR 2458259 |
| Reference:
|
[27] Mro.zek N.: Ideal version of Egoroff's theorem for analytic $P$-ideals.J. Math. Anal. Appl. 349 (2009), 452–458. MR 2456202, 10.1016/j.jmaa.2008.08.032 |
| Reference:
|
[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. Zbl 1221.28012, MR 1958294 |
| Reference:
|
[29] Šalát T.: On statistically convergent sequences of real numbers.Math. Slovaca 30 (1980), 139–150. Zbl 0437.40003, MR 0587239 |
| Reference:
|
[30] Schoenberg I.J.: The integrability of certain functions and related summability methods.Amer. Math. Monthly 66 (1959), 361–375. Zbl 0089.04002, MR 0104946, 10.2307/2308747 |
| Reference:
|
[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. MR 0776622 |
| . |
