![MathJax Logo](/templates/jsp/_style2/_tandf/pb2/images/math-jax.gif)
Abstract
Bourbaki complete metric spaces are important since they are a class between compact metric spaces and complete metric spaces. The aim of the present paper is to introduce the statistical Bourbaki–Cauchy sequence as a new concept and to give an equivalent condition for a metric space to be Bourbaki complete. Also, Bourbaki complete and Bourbaki-bounded metric spaces are characterized in terms of functions which preserve statistical Bourbaki–Cauchy sequences.
PUBLIC INTEREST STATEMENT
Compact metric spaces and complete metric spaces play an important role in functional analysis. Metric spaces satisfying properties between compactness and completeness have been the subject of research for a number of papers over years. Bourbaki complete metric spaces in which every Bourbaki–Cauchy sequence clusters are complete but not necessarily compact. Bourbaki–Cauchy sequences are defined recently to characterize Bourbaki-bounded metric spaces as Cauchy sequences characterize totally bounded metric spaces. In this paper, some new characterizations of Bourbaki-bounded and Bourbaki complete metric spaces are given.
1. Introduction
Throughout this paper, and
will stand for the set of all natural numbers and real numbers, respectively.
Complete metric spaces play an important role in logic, fixed point theory, computer science, quantum mechanics and other branches of science as well as in functional analysis. Also, compactness is central to the theory of metric spaces. It is a well-known fact that a continuous function from a compact metric space to any metric space is uniformly continuous whereas compactness is not necessary. For instance, a continuous function defined on a uniformly discrete metric space with infinitely many points is uniformly continuous, but this discrete space is not compact. Since every compact metric space is complete (whereas the reverse is not the case), metric spaces satisfying properties stronger than completeness but weaker than compactness have been investigated by many mathematicians. The most known of such spaces is the Atsuji space (also called UC space) defined as every real-valued continuous function on it is uniformly continuous. Because of the importance of such space, many authors gave some different characterizations of this space. First, Nagata (Nagata, Citation1950) studied on Atsuji spaces. Later, in (Atsuji, Citation1958; Monteiro & Peixoto, Citation1951), the authors gave many new equivalent conditions for a metric space to be an Atsuji space. In a survey article by Kundu and Jain (Kundu & Jain, Citation2006), twenty-five equivalent conditions are brought close together. Further, Beer in the papers (Beer, Citation1985, Citation1986) investigated Atsuji spaces. Most recently, some new practical and exotic characterizations of these spaces are presented in a different aspect by Aggarwal and Kundu (Aggarwal & Kundu, Citation2016). For more papers about Atsuji spaces, one can see Aggarwal & Kundu, (Citation2017); Jain & Kundu, (Citation2007).
For a metric space, features which remain in between compactness and completeness are studied by many authors. One of the nice papers related to this subject is (Beer, Citation2008) by Beer. As well as being an Atsuji space, being a boundedly compact, a uniformly locally compact, a cofinally complete or a strongly cofinally complete space can be given as examples of such features. Recently, a new pair of these features are presented by Garrido and Meroño (Garrido & Meroño, Citation2014) such a way that clustering all sequences belonging to a more general class than the class of Cauchy sequences to make this property stronger than completeness. Hence, this property is weaker than compactness since every sequence has a convergent subsequence in a compact metric space. First, they define the concept of a Bourbaki–Cauchy sequence which is more general than a Cauchy sequence. A sequence in a metric space
is said to be Bourbaki–Cauchy if for every
there exist
and
such that
for
, where
consists of points
satisfying
for some
. Then, it is obvious that every Cauchy sequence is a Bourbaki–Cauchy sequence (but not reverse). Unlike a complete metric space, Bourbaki completeness is defined as every Bourbaki–Cauchy sequence in
has a convergent subsequence. Since a Bourbaki–Cauchy sequence may have more than one cluster point, the sequence itself cannot be convergent. As an example, the sequence
in
with the usual metric is a Bourbaki–Cauchy sequence which is not Cauchy and so not convergent but has some convergent subsequences. Second, they define a cofinally Bourbaki–Cauchy sequence and a cofinally Bourbaki complete metric space analogous with Bourbaki complete metric space. The class of Bourbaki–Cauchy sequences and cofinally Bourbaki–Cauchy sequences appeared to characterize a Bourbaki-bounded subset of a metric space in a similar way that a Cauchy sequence characterizes total boundedness of a set. For the first time, Atsuji (Atsuji, Citation1958) introduced this concept of boundedness under the name of finitely chainable to study metric spaces on which every real-valued uniformly continuous function is bounded. A subset
of a metric space
is said to be Bourbaki bounded if for every
there exist
and finitely many points
such that
. In a metric space, a totally bounded set is Bourbaki bounded and a Bourbaki-bounded set is bounded in the usual sense.
As an extension of usual convergence, the concept of statistical convergence for real-valued sequences was introduced by Fast (Fast, Citation1951) and Steinhaus (Steinhaus, Citation1951). However, the idea of statistical convergence (appeared under the name of almost convergence) goes back to Zygmund (Zygmund, Citation2002) (first edition published in Warsaw 1935). The formal definition is based on the notion of natural density (asymptotic density) of a subset in
(Niven, Zuckerman, & Montgomery, Citation1991). If the limit
exists, it is called as the natural density of
and denoted by
, where
is the characteristic function of
(that is,
if
; else
). Along the paper, when
appears, we mean that it is well defined. Also, note that the following statements are true for any subsets
in
.
(1) If exists, then
and
also exists with
.
(2) If and
, then
.
(3) If and
, then
.
(4) If and
, then
.
The statistical convergence was generalized to sequences in some other spaces and studied on these spaces. For example, it has been considered in metric spaces (Küçükaslan, Değer, & Dovgoshey, Citation2014), cone metric spaces (Li, Lin, & Ge, Citation2015), topological and uniform spaces (Di Maio & Kočinac, Citation2008) and topological groups (Çakall, Citation2009). In Schoenberg (Citation1959), Schoenberg gave some basic properties of statistical convergence and also studied the concept as a summability method. Later on it was further investigated and linked with the summability theory by Fridy (Fridy, Citation1985), Fridy and Orhan (Fridy & Orhan, Citation1993), Mursaleen and Edely (Mursaleen & Edely, Citation2004), Acar and Mohiuddine (Acar & Mohiuddine, Citation2016), M. Aldhaifallah et al (Aldhaifallah, Nisar, Srivastava, & Mursaleen, Citation2017), Belen and Mohiuddine (Belen & Mohiuddine, Citation2013), Kirici and Karaisa (Citation2017), Braha et al (Braha, Srivastava, & Mohiuddine, Citation2014) and many others. Also, several important applications of statistical convergence is available in different areas of mathematics such as measure theory (Miller, Citation1995), optimization theory (Pehlivan & Mamedov, Citation2000), approximation theory (Edely, Mohiuddine, & Noman, Citation2010, Gadjiev & Orhan, Citation2002, Kadak, Braha, & Srivastava, Citation2017, Kadak & Mohiuddine, Citation2018, Srivastava, Jena, Paikray, & Mishra, Citation2018), probability theory (Fridy & Khan, Citation1998), etc. A sequence
in a metric space
statistically converges to a point
if for every
we have
, where
. A sequence
is a statistical Cauchy sequence in
if for every
there exists
such that
, where
. Also,
is said to be statistically bounded in
if there exist
and
such that
.
In this paper, we define the statistical Bourbaki–Cauchy sequence as a new concept in the setting of metric spaces. By their definitions, being a Bourbaki–Cauchy sequence or a statistical Cauchy sequence implies that this sequence is also a statistical Bourbaki–Cauchy sequence. However, a statistical Bourbaki–Cauchy sequence need not be Bourbaki–Cauchy or statistical Cauchy which can be seen in Example 2.2 and Example 2.3. Further, we state a new uniform condition by the aid of a statistical Bourbaki Cauchy sequence and prove that it is equivalent to Bourbaki completeness (see Theorem 2.6). Moreover, we study some new characterizations of Bourbaki completeness and Bourbaki boundedness of a metric space by using functions which preserve statistical Bourbaki Cauchy sequences.
2. Statistical Bourbaki–Cauchy sequence and some results related to this concept
We start this section with the definition of a statistical Bourbaki–Cauchy sequence by using the concept of natural density of a set in . Later on, we examine the relations between this new sequence with some other sequences defined earlier in the literature.
Definition 2.1. A sequence in a metric space
is said to be statistical Bourbaki–Cauchy if for every
there exist
and
such that
.
Equivalently, this definition can be given as for every there exist
and
such that
owing to the fact that
for all subset
of
.
From the definitions, it is clear that every Bourbaki–Cauchy sequence in a metric space is also statistical Bourbaki–Cauchy. But the reverse implication is not true as the following example shows. One of the most interesting difference between these two type of Cauchy sequences is that statistical Bourbaki–Cauchy sequences are not generally bounded whereas Bourbaki–Cauchy sequences are bounded in the sense of the metric. On the other hand, if is a statistical Bourbaki–Cauchy sequence in a metric space
, then given any
, we have
for some and
which shows that
is statistically bounded.
Example 2.2. Consider with the usual metric. The sequence
defined in the following way
is in fact statistical Cauchy and so statistical Bourbaki–Cauchy due to the fact that the natural density of the set of all prime numbers equals to zero (see Kováč, Citation2005). However, it is not a Bourbaki–Cauchy sequence since it is not bounded with respect to the usual metric.
Obviously, statistical convergence of a sequence in a metric space implies that the sequence is a statistical Bourbaki–Cauchy sequence. But there are statistical Bourbaki–Cauchy sequences in some metric spaces which are not statistically convergent such as given in the following example.
Example 2.3. The sequence in
with the usual metric is a Bourbaki–Cauchy sequence and therefore it is a statistical Bourbaki–Cauchy sequence. But it is not statistical Cauchy. Although it has statistically convergent subsequences, the sequence
itself is not statistically convergent.
These last two examples show that if a statistical Bourbaki–Cauchy sequence has a statistically convergent subsequence, then the sequence itself does not have to be statistically convergent likewise a Bourbaki–Cauchy sequence. Also, it can be seen that there is no relation between statistically Cauchy and Bourbaki–Cauchy sequences. Consequently, we have the following diagram where the reverse implications do not hold.
In the following theorem, some relations between Bourbaki–Cauchy and statistical Bourbaki–Cauchy sequences are obtained.
Theorem 2.4. For a sequence in a metric space
, the following statements are equivalent.
(1) is a statistical Bourbaki–Cauchy sequence in
.
(2). There exists a Bourbaki–Cauchy subsequence of
such that
.
(3). There exists a statistical Bourbaki–Cauchy subsequence of
such that
.
Proof. Let
be a statistical Bourbaki–Cauchy sequence in
. Then, there exist
and
such that
, where
. Similarly, there exist
and
such that
, where
. Put
. Then, we have
,
and
for all
. By continuing this process, we obtain a decreasing sequence
of subsets of
with
and
for all
. Let
and choose
with
such that
for all
. In this manner, we construct an increasing sequence
in
such that
for all
, where
for each
. Set
. For any
and
, we have
which implies that
. Now, given any
, we can find a natural number
satisfying
. Choose fixed
and an arbitrary
with
. Then, there exist
with
such that
,
and
,
. Hence, we have
and so
which means that
is the desired Bourbaki–Cauchy subsequence.
The implication follows from the fact that a Bourbaki–Cauchy sequence is a statistical Bourbaki–Cauchy sequence.
Let
be a statistical Bourbaki–Cauchy subsequence of
, where
. Then, given any
there exist
and
such that
where and
, respectively. We conclude that
which proves that the sequence
is statistical Bourbaki–Cauchy in
. □
As a consequence of this theorem, we have the following result.
Corollary 2.5. Every statistical Bourbaki–Cauchy sequence has a Bourbaki–Cauchy subsequence in a metric space.
The next result states a condition which is equivalent to Bourbaki completeness.
Theorem 2.6. A metric space is Bourbaki complete if and only if every statistical Bourbaki–Cauchy sequence has a statistical convergent subsequence.
Proof. Suppose that a metric space is Bourbaki complete. Let
be a statistical Bourbaki–Cauchy sequence in
. By the previous theorem, it has a Bourbaki–Cauchy subsequence. Then, Bourbaki completeness of
implies that it has a usual convergent and so statistical convergent subsequence.
For the converse, take a Bourbaki–Cauchy sequence in
. Since it is also statistical Bourbaki–Cauchy, by hypothesis there exists a statistical convergent subsequence of
. Since every statistical convergent sequence has a convergent subsequence (see Lemma 1.1 in Šalát, Citation1980), it follows that
is a Bourbaki complete metric space. □
In a recent paper (Kundu, Aggarwal, & Hazra, Citation2017), Kundu et al. studied three new characterizations of Bourbaki-bounded metric spaces. For this purpose, they used various types of functions defined in (Aggarwal & Kundu, Citation2017). One of them is a Bourbaki–Cauchy regular function required for our main results. A function is said to be Bourbaki–Cauchy regular if
is a Bourbaki–Cauchy sequence in
whenever
is a Bourbaki–Cauchy sequence in
. Also, we need to recall some more definitions. In a metric space,
, for
, the ordered set
in
is called an
-chain of length
from
to
if
holds for
. It is said that
is
-chainable if each two points of
can be joined by an
-chain, and
is chainable if
is
-chainable for every
.
In the next theorem, some new characterizations of Bourbaki completeness are given by using functions which preserve statistical Bourbaki–Cauchy sequences and can be named as a statistical Bourbaki–Cauchy regular function. Further, in the last theorem, Bourbaki boundedness of a metric space is characterized in terms of these functions. Before characterizing Bourbaki completeness, we examine the relation between statistical Bourbaki–Cauchy regular and Bourbaki–Cauchy regular functions.
Lemma 2.7. Each Bourbaki–Cauchy regular function is statistical Bourbaki–Cauchy regular.
Proof. Let be a Bourbaki–Cauchy regular function and
be a statistical Bourbaki–Cauchy sequence in
. Then, by Theorem 2.4, it has a Bourbaki–Cauchy subsequence
such that
. Hence, our assumption implies that the sequence
is also Bourbaki–Cauchy. It follows again from Theorem 2.4 that the sequence
is statistical Bourbaki–Cauchy. Thus, we conclude that the function
is statistical Bourbaki–Cauchy regular.
Hereby, we give one of our main results.
Theorem 2.8. The following statements are equivalent for a metric space .
(1) is Bourbaki complete.
(2) Every continuous function from into a chainable metric space
is Bourbaki–Cauchy regular.
(3). Every continuous function from into a chainable metric space
is statistical Bourbaki–Cauchy regular.
(4) Every continuous function from into
is statistical Bourbaki–Cauchy regular.
Proof. It is proved in [3, Theorem 2.4].
The proof comes from the fact that every Bourbaki–Cauchy regular function is statistical Bourbaki–Cauchy regular which is proved in Lemma 2.7.
It is clear since
is chainable with respect to the usual metric.
Let
be a Bourbaki–Cauchy sequence in
. We can say that
has a Bourbaki–Cauchy subsequence whose terms are distinct; otherwise, there is nothing to prove. Now, suppose that a Bourbaki–Cauchy sequence
with distinct terms has no convergent subsequence. It follows that
is a closed subset of
. Also, the subspace topology on the set
is discrete topology since it consists of only isolated points. Define a real-valued function
on
with
for all
. Then
is a continuous function since every function defined on a discrete topological space is continuous. Accordingly, Tietze extension theorem implies that there is a continuous function
with
for all
. But this function cannot be statistical Bourbaki–Cauchy regular since the sequence
is not a statistical Bourbaki–Cauchy sequence in
whereas
is statistical Bourbaki–Cauchy in
. Therefore, every Bourbaki–Cauchy sequence in
must have a convergent subsequence which means that
is Bourbaki complete. □
Theorem 2.9. The following statements are equivalent for a metric space .
(1) Every sequence in has a statistical Bourbaki–Cauchy subsequence.
(2) If is a statistical Bourbaki–Cauchy regular function, where
is any metric space, then
is bounded.
(3). If is a statistical Bourbaki–Cauchy regular function, where
is an unbounded chainable metric space, then
is bounded.
(4) is Bourbaki bounded.
Proof. Suppose that
is a statistical Bourbaki–Cauchy regular function but not a bounded function. Then for all
, we can construct a sequence
in
satisfying
since the set
is not bounded. By hypothesis, the sequence
has a statistical Bourbaki–Cauchy subsequence, say
. However,
is not a statistical Bourbaki–Cauchy sequence. Indeed, given any
and
, the set
is finite. Otherwise, for a fixed
, the inclusion
implies that for infinitely many
. Hence,
is not a statistical Bourbaki–Cauchy sequence which contradicts the fact that
is statistical Bourbaki–Cauchy regular. Thus,
must be a bounded function.
This is obvious.
Suppose that
is not Bourbaki bounded. Then, there exists an
such that for all
,
cannot be covered by a union of finitely many sets
. Fix
. Then, we can choose
such that
. In the same manner, we can choose
such that
. By continuing this process, we obtain a sequence
in
such that
for every
and
. Let
. Since
is an unbounded metric space, there is a point
such that
for all
. By virtue of this fact, we define an unbounded function
as:
However, this function is statistical Bourbaki–Cauchy regular. To observe this, take a statistical Bourbaki–Cauchy sequence in
. Then for this
, there exit a natural number
and a point
such that
; that is
contains infinitely many terms of the sequence
. On the other hand, for only finitely many
,
, where
. Otherwise, since the inclusion
holds for infinitely many , we contradict with the construction of the sequence
. Hence,
is a finite subset of
. It follows that given any
,
, where
and
is the length of the
-chain from
to
for every
. Thus, we conclude that the subsequence
is Bourbaki–Cauchy with
which means the sequence itself
is a statistical Bourbaki–Cauchy sequence in
. Consequently, we obtain an unbounded statistical Bourbaki–Cauchy regular function from
into unbounded chainable metric space
opposite to hypothesis and so
is Bourbaki bounded.
It is proved in [18, Theorem 4] that if
is Bourbaki bounded, then every sequence in
has a Bourbaki–Cauchy subsequence and so it has a statistical Bourbaki–Cauchy subsequence.
3. Conclusion
Compact metric spaces and complete metric spaces with their basic properties are well known by all mathematicians and metric spaces satisfying properties between compactness and completeness have been the subject of research for many papers over years. One such well-known metric space is Atsuji or UC space on which every real-valued continuous function is uniformly continuous. Also, a Bourbaki complete metric space can be given as an example of such an intermediate property defined and studied in the recent time. It has been proved that every UC metric space is Bourbaki complete. In this present paper, we state a new condition equivalent to Bourbaki completeness by defining a new class of sequences named as a statistical Bourbaki–Cauchy sequence. Hence, we conclude that every sequence in any UC space has a statistical Bourbaki–Cauchy subsequence. Further, since compactness has been characterized by Bourbaki boundedness and Bourbaki completeness, we can say that a metric space is compact if and only if
is Bourbaki bounded and every sequence in
has a statistical Bourbaki–Cauchy subsequence.
Additional information
Funding
Notes on contributors
Merve Ilkhan
Merve I˙lkhan received his BSc (Mathematics) (2012) degree from Istanbul Commerce University, Turkey and MSc (Mathematics) (2014) and PhD (Mathematics) (2018) from Duzce University, Turkey. She is interested in Functional Analysis, Topology, Summability Theory, Sequence Spaces, Measure of Noncompactness and Operator Theory. This paper is a part of her PhD thesis.
Emrah Evren Kara
Emrah Evren Kara received his BSc (Mathematics), MSc (Mathematics) and PhD (Mathematics) degree from Sakarya University, Turkey in 2006, 2008 and 2012, respectively. Moreover, he is founder and Editor-in-Chief of Universal Journal of Mathematics and Applications, Journal of Mathematical Sciences and Modelling and Managing Editor of Fundamental Journal of Mathematics and Applications. His main research interests are: Sequence Spaces, Summability Theory, Measure of Noncompactness, Operator Theory. He has published many research papers in reputed international journals.
References
- Acar, T., & Mohiuddine, S. A. (2016). Statistical (C,1)(E,1) summability and Korovkin’s theorem. Filomat, 30(2), 361-375. doi:10.2298/FIL1602387A
- Aggarwal, M., & Kundu, S. (2016). More on variants of complete metric spaces. Acta Mathematica Hungarica. doi:10.1007/s10474-016-0682-2
- Aggarwal, M., & Kundu, S. (2017). Boundedness of the relatives of uniformly continuous functions. Topology Proceedings, 49, 105–119.
- Aldhaifallah, M., Nisar, K. S., Srivastava, H. M., & Mursaleen, M. (2017). Statistical Λ-convergence in probabilistic normed spaces. Applied Mathematics and Computation, 2017, 7.
- Atsuji, M. (1958). Uniform continuity of continuous functions of metric spaces. Pacific Journal of Mathematics, 8, 11–16. doi:10.2140/pjm
- Beer, G. (1985). Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance. Proceedings of the American Mathematical Society, 95, 653–658. doi:10.1090/S0002-9939-1985-0810180-3
- Beer, G. (1986). More about metric spaces on which continuous functions are uniformly continuous. Bulletin of the Australian Mathematical Society, 33, 397–406. doi:10.1017/S0004972700003981
- Beer, G. (2008). Between compactness and completeness. Topology and Its Applications, 155, 503–514. doi:10.1016/j.topol.2007.08.020
- Belen, C., & Mohiuddine, S. A. (2013). Generalized weighted statistical convergence and application. Applied Mathematics and Computation, 219(18), 9821–9826.
- Braha, N. L., Srivastava, H. M., & Mohiuddine, S. A. (2014). A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallee Poussin mean. Applied Mathematics and Computation, 228, 162–169.
- Çakall, H. (2009). A study on statistical convergence. Functional Analysis, Approximation and Computation, 1(2), 19–24.
- Di Maio, G., & Kočinac, L. D. R. (2008). Statistical convergence in topology. Topology and Its Applications, 156, 28–45. doi:10.1016/j.topol.2008.01.015
- Edely, O. H., Mohiuddine, S. A., & Noman, A. K. (2010). Korovkin type approximation theorems obtained through generalized statistical convergence. Applied Mathematics Letters, 23(11), 1382–1387. doi:10.1016/j.aml.2010.07.004
- Fast, H. (1951). Sur la convergence statistique. Colloquium Mathematicum, 2, 241–244. doi:10.4064/cm-2-3-4-241-244
- Fridy, J. A. (1985). On statistical convergence. Analysis, 5, 301–313. doi:10.1524/anly.1985.5.4.301
- Fridy, J. A., & Khan, M. K. (1998). Tauberian theorems via statistical convergence. Journal of Mathematical Analysis and Applications, 228, 73–95. doi:10.1006/jmaa.1998.6118
- Fridy, J. A., & Orhan, C. (1993). Lacunary statistical summability. Journal of Mathematical Analysis and Applications, 173, 497–504. doi:10.1006/jmaa.1993.1082
- Gadjiev, A. D., & Orhan, C. (2002). Some approximation theorems via statistical convergence. Rocky Mountain Journal of Mathematics, 32, 129–138. doi:10.1216/rmjm/1030539612
- Garrido, M. I., & Meroño, A. S. (2014). New types of completeness in metric spaces. Annales Academiae Scientiarum Fennicae Mathematica, 39, 733–758. doi:10.5186/aasfm.00
- Jain, T., & Kundu, S. (2007). Atsuji completions: Equivalent characterisations. Topology and Its Applications, 154, 28–38. doi:10.1016/j.topol.2006.03.014
- Kadak, U., Braha, N. L., & Srivastava, H. M. (2017). Statistical weighted B-summability and its applications to approximation theorems. Applied Mathematics and Computation, 302, 80–96.
- Kadak, U., & Mohiuddine, S. A. (2018). Generalized statistically almost convergence based on the difference operator which includes the (p,q)-gamma function and related approximation theorems. Results Mathematical, 73(9). doi:10.1007/s00025-018-0789-6
- Kirişci, M., & Karaisa, A. (2017). Fibonacci statistical convergence and Korovkin type approximation theorems. Journal of Inequalities and Applications, 2017(229), 15.
- Kováč, E. (2005). On φ-convergence and φ-density. Mathematical Slovaca, 55(3), 329–351.
- Küçükaslan, M., Değer, U., & Dovgoshey, O. (2014). On the statistical convergence of metric-valued sequences. Ukrainian Mathematical Journal, 66(5), 796–805. doi:10.1007/s11253-014-0974-z
- Kundu, S., Aggarwal, M., & Hazra, S. (2017). Finitely chainable and totally bounded metric spaces: Equivalent characterizations. Topology and Its Applications, 216, 59–73. doi:10.1016/j.topol.2016.11.008
- Kundu, S., & Jain, T. (2006). Atsuji spaces: Equivalent conditions. Topology Proceedings, 30, 301–325.
- Li, K., Lin, S., & Ge, Y. (2015). On statistical convergence in cone metric spaces. Topology and Its Applications, 196, 641–651. doi:10.1016/j.topol.2015.05.038
- Miller, H. I. (1995). A measure theoretical subsequence characterization of statistical convergence. Transactions of the American Mathematical Society, 347, 1811–1819.
- Monteiro, A. A., & Peixoto, M. M. (1951). Le nombre de Lebesgue et la continuité uniforme. Portugaliae Mathematical, 10, 105–113.
- Mursaleen, M., & Edely, O. H. H. (2004). Generalized statistical convergence. Information Sciences, 162, 287–294. doi:10.1016/j.ins.2003.09.011
- Nagata, J. (1950). On the uniform topology of bicompactifications. Journal of the Institute of Polytechnics, Osaka City University, 1, 409–422.
- Niven, I., Zuckerman, H. S., & Montgomery, H. (1991). An introduction to the theory of numbers (5th ed.). New York, NY: Wiley.
- Pehlivan, S., & Mamedov, M. A. (2000). Statistical cluster points and turnpike. Optimization, 48, 93–106. doi:10.1080/02331930008844495
- Šalát, T. (1980). On statistically convergent sequences of real numbers. Mathematical Slovaca, 30(2), 139–150.
- Schoenberg, I. J. (1959). The integrability of certain functions and related summability methods. The American Mathematical Monthly, 66, 361–775. doi:10.1080/00029890.1959.11989303
- Srivastava, H. M., Jena, B. B., Paikray, S. K., & Mishra, U. K. (2018). A certain class of weighted statistical convergence and associated Krovkin-type approximation theorems involving trigonometric functions. Mathematical Methods Applications Sciences, 41, 671–683.
- Steinhaus, H. (1951). Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematics, 2, 73–74.
- Zygmund, A. (2002). Trigonometric series (3rd ed.). Cambridge, UK: Cambridge University Press.