On summability of nets through ideals and some topological observations
by
Pratulananda Das
Jadavpur University, Kolkata -700032, West Bengal, India
Coauthors: Sanjoy Kr Ghoshal

The idea of convergence of a real sequence had been extended to statistical convergence by Fast [2] ( see also Schoenberg [13] ) as follows: If N denotes the set of natural numbers and K ⊂ N then K_n denotes the set {k ∈ K: k ≤ n} and |K_n| stands for the cardinality of the set K_n. The natural density of the subset K is defined by

d(K)= lim n→∞  |Kn| n

provided the limit exists.

A sequence {x_n}_{n ∈ N} of points in a metric space (X, r) is to be statistically convergent to l if for arbitrary e > 0, the set K(e)={k ∈ N :d(x_k, l) ≥ e} has natural density zero. A lot of investigations have been done on this convergence and its topological consequences after the initial works by Fridy [3] and Salát [12]. In particular, Very recently Di maio and Kocinak [8] introduced the concept of statistical convergence in topological spaces as well as uniform spaces and established the topological nature of this convergence as also offered some applications to selection principles theory, function spaces and hyper spaces.

However if one considers the concept of nets instead of sequences ( which undoubtedly play more important and natural role in topological and uniform spaces ) the above approach does not seem to be appropriate because of the absence of any idea of density in arbitrary directed sets. Instead it seems more appropriate to follow the more general approach of [4].

In [4] an interesting generalization of the notion of statistical convergence was proposed. Namely it is easy to check that the family I_d = {A ⊂ N:d(A)=0 } forms a non-trivial admissible ideal of N ( recall that I ⊂ 2^N is called an ideal if (i) f ∈ I, (ii) A, B ∈ I implies A ∪B ∈ I and (iii) A ∈ I, B ⊂ A implies B ∈ I. I is called non-trivial if I ≠ {f} and N ∉ I . I is admissible if it contains all singletons ). Thus one may consider an arbitrary ideal I of N and define I-convergence of a sequence as follows.

A sequence {x_n}_{n ∈ N} in (X, r ) is said to be I-convergent to x ∈ X, ( in short x=I-lim_n→∞x_n ) if K(e) ∈ I for each e > 0, where K(e)={k ∈ N:d(x_k, x) ≥ e}.

The aim of this presentation is to show that the idea of convergence and Cauchy condition of nets can be broadened in the same way using the concept of ideals. It is observed that two types of convergence namely, I-convergence and I^*-convergence can be considered in line of statistical and s^*-convergence of [8] as well as the corresponding Cauchy conditions. But unlike [8], these concepts are not in general equivalent even in first countable spaces ( which can be shown by constructing proper examples ) and only coincide if and only if the ideal satisfies a condition called condition (DP). The basic topological nature of these convergence are established and most importantly an open problem posed by Di Maio and Kocinac (Problem 2.16 [8]) is considered and we try to give some answers.This motivates us to consider the idea of completeness of an uniform space and we also consider the impact of this generalization on the notion of completeness and make certain interesting observations. Finally a kind of divergence of nets is considered in uniform spaces and its basic properties are studied.

References:

[1] K.Dems, On I-Cauchy sequences, Real Anal. Exchange, 30 (1) (2004-2005), 123-128.
[2] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
[3] J.A.Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.
[4] P.Kostyrko, T.Salát, W.Wilczynki, I-convergence, Real Anal. Exchange, 26 (2) (2000/2001), 669-685.
[5] K.Kuratowski, Topologie I, PWN, Warszawa, 1961.
[6] B.K.Lahiri, Pratulananda Das, I and I^* convergence in topological spaces, Math. Bohemica, 130 (2) (2005), 153-160.
[7] B.K.Lahiri, Pratulananda Das, I and I^* convergence of nets, Real Anal. Exchange, 33 (2) (2007-2008), 431 - 442.
[8] G. Di Maio, Lj. D.R. Kocinac, Statistical convergence in topology, Top. Appl., 156 (2008), 28 - 45.
[9] A. Nabiev, S. Pehlivan, M. Gurdal, On I-Cauchy sequences, Taiwanese J. Math., 11 (2), (2007), 569 - 576.
[10] Pratulananda Das, S. Ghoshal, On I-Cauchy nets and completeness, communicated to.
[11] Pratulananda Das, S Ghoshal, On some further remarks on ideal summability of nets in uniform spaces, under preparation.
[12] T. Salát, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), 139-150.
[13] I.J.Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.

Date received: October 6, 2009