ON CONJECTURE 2 OF NADA AND ZOHNY FROM THE PERSPECTIVE OF BITOPOLOGICAL DYNAMICAL SYSTEMS

Bitopological dynamical system is a recently explored area of dynamical system to investigate dynamical properties in terms of a bitopological space. Nada and Zohny [S.I. Nada, H. Zohny, An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202–204] explored the use of topological dynamical system in the development of an organism from zygote until birth and they made three conjectures regarding the development of child growth from zygote to till its birth. In this paper, we disprove the conjecture 2 of Nada and Zohny [S.I. Nada, H. Zohny, An application of relative topology in biology, Chaos, Solitons and Fractals. 42 (2009), 202–204] by applying some mathematical results from bitopological dynamical system, which was recently introduced by Acharjee et al. [S. Acharjee, K. Goswami, H.K. Sarmah, Transitive map in bitopological dynamical systems, (communicated)] with medical evidences. Also, we introduce forward iterated connected space, backward iterated connected space, pairwise iterated connected space and establish some of their relations with pairwise connectedness in bitopological dynamical system. We show that during the development of an organism from zygote until birth, the developing stage after gastrulation is pairwise disconnected and forward iterated disconnected.


INTRODUCTION
Bitopological dynamical system is a recently explored area of dynamical system to investigate dynamical properties in terms of a bitopological space. Because of its applications in human embryo development, it shows the potential to be a promising area of dynamical system.
Bitopological dynamical system, introduced by Acharjee et al. [1], overcomes the limitations of topological dynamical system. In [1], authors disproved the conjecture 1 of Nada and Zohny [2] by using results of bitopological dynamical system and introduced some new notions like transitive map, pairwise iterated compactness, etc. Moreover, in another paper [3], they introduced forward iterated Hausdorffness and showed that the developing stage after gastrulation is forward iterated Hausdorff in the development of an organism from zygote until birth. It motivated us for this paper on viewing the possibilities to use bitopological dynamical system to study the developmental anomalies of human brain.
Kelly [4] introduced the concept of bitopological space. Later, bitopological space was extensively studied by many researchers of various branches and the study is still going on. For recent theoretical works in bitopological space, one may refer to Acharjee and Tripathy [5], Acharjee et al. [6], Acharjee et al. [7] and many others. Recently, bitopological space has been applied in many areas of science and social science, viz. medical science [8], economics ([9], [10]), computer science [11], human embryo development ( [1], [3]), etc. Moreover, entropy of pairwise continuous map in bitopological dynamical system was recently introduced by Acharjee et al. [29].
Reilly [12] introduced the concept of pairwise connectedness in a bitopological space. After that many researchers studied the concept of pairwise connectedness because of its fundamental importance in the theory of bitopological space. Recently, Mishra and Singh [13] introduced the concept of pairwise connectedness for hyperspace of a bitopological space (briefly, called bihypertopological space).
Mutation is a change in the DNA sequence of a gene. Many researchers ( [14], [15], [16]) have studied the mutations of certain genes as causes of various anomalies of brain development.
This paper is divided into three sections. In the preliminary section, we recall some existing definitions of bitopological space as well as bitopological dynamical system. In the next section, we introduce forward iterated connected space, backward iterated connected space and pairwise iterated connected space; which generalize the notion of pairwise connectedness in a bitopological dynamical system. Also, we establish some relationships among these generalizations.
Then, we show that the developing stage after gastrulation is forward iterated disconnected in the development of an organism from zygote until birth. Finally, in the last section we disprove the conjecture 2 of Nada and Zohny [2] by using the concept of pairwise Hausdorffness and gene mutation. Our theory is supported by medical evidences.

PRELIMINARY DEFINITIONS
In this section, we recall some existing definitions of bitopological space and bitopological dynamical system.

Definition 2.1.[4]
A quasi-pseudo-metric on a set X is a non-negative real-valued function p(, ) on the product X × X such that (ii) p(x, z) ≤ p(x, y) + p(y, z), where x, y, z ∈ X.
Definition 2.2. [4] Let p(, ) be a quasi-pseudo-metric on X, and let q(, ) be defined by q(x, y) = p(y, x), where x, y ∈ X. Then, q(, ) is also a quasi-pseudo-metric on X. We say that p(, ) and q(, ) are conjugate, and denote the set X with the structure by (X, p, q).
If p(, ) is a quasi-pseudo-metric on a set X, then the open p-sphere with centre x and radius ε > 0 is the set S p (x, ε) = {y : p(x, y) < ε}. The collection of all open p-spheres forms a base for a topology. Similarly, q(, ) determines a topology for X. We shall denote the topology determined by p(, ) by τ 1 and the topology that of q(, ) by τ 2 .
Pervin [17] called this a continuous map. However, we call this as pairwise continuous map, due to Reilly [12].
Definition 2.5. [12] A bitopological space (X, τ 1 , τ 2 ) is pairwise connected iff X cannot be expressed as the union of two non-empty disjoint sets A and B such that (A ∩ τ 1 -cl B) ∪ (τ 2cl A ∩ B) = φ ; where τ 1 -cl A denotes the τ 1 -closure of the set A. If X can be so expressed, we write X = A|B and this is a separation of X, which is then pairwise disconnected.
Definition 2.6. [4] A bitopological space (X, τ 1 , τ 2 ) is said to be pairwise Hausdorff if, for each two distinct points x and y, there are a τ 1 -neighbourhood U of x and a τ 2 -neighbourhood V of y In [1], we considered N, Z and R as the set of non-negative integers, the set of integers and the set of real numbers, respectively.
Definition 2.10. [3] Let (X, f ) be a bitopological dynamical system, where (X, τ 1 , τ 2 ) is a bitopological space and f : X → X is a pairwise continuous map. We call (X, f ) as (m, n)-forward iterated Hausdorff if for each two distinct points x and y of X, there exist m, n ∈ N, U ∈ τ 1 and Theorem 2.1. [17] The following conditions are equivalent for any bitopological space (X, τ 1 , τ 2 ): (a) (X, τ 1 , τ 2 ) is connected, (b) X cannot be expressed as the union of two non-empty disjoint sets A and B such that A is

MAIN RESULTS
In this section, our main aim is to introduce forward iterated connectedness, backward iterated connectedness, pairwise iterated connectedness and some related results. Moreover, we show that the developing stage after gastrulation is forward iterated disconnected in the development of an embryo from zygote until birth.
Let N, Z and R denote the set of non-negative integers, the set of integers and the set of real numbers, respectively.
Pairwise connectedness is a basic concept in bitopological space. So, in next few definitions we try to define some generalized versions of pairwise connectedness in bitopological dynamical system with respect to iteration. Definition 3.1. Let (X, f ) be a bitopological dynamical system, where (X, τ 1 , τ 2 ) is a bitopological space and f : X → X is a pairwise continuous map. We call (X, f ) as forward iter- Let (X, f ) be a bitopological dynamical system, where (X, τ 1 , τ 2 ) is a bitopological space and f : X → X is a pairwise continuous map. We call (X, f ) as pairwise iter- The following theorem establishes the relation between pairwise connectedness, forward iterated connectedness, backward iterated connectedness and pairwise iterated connectedness without considering any extra condition.
connected or backward iterated connected or pairwise iterated connected, then X is pairwise connected. Now, we get the following relation between pairwise connectedness and forward iterated connectedness in case of a +invariant map.
Proof: Since X is pairwise connected, so X cannot be expressed as the union of two non- Thus, for all m, n ∈ N, U( = φ ) ∈ τ 1 and V ( = φ ) ∈ τ 2 ; For a −invariant map, we get the following result.
Proof: The pairwise connectedness of the bitopological space X implies that X cannot be expressed as the union of two non-empty disjoint sets U and V such that U is τ 1 -open and V is τ 2open. Let us consider m, n ∈ N be arbitrary. Now, the pairwise continuity of the map f implies Thus, for all m, n ∈ N, Theorem 3.4. Let (X, f ) be a bitopological dynamical system, where X is pairwise connected.
If f is invariant, then (X, f ) is pairwise iterated connected.
Proof: Since X is pairwise connected, so X cannot be expressed as the union of two non-empty disjoint sets U and V such that U is τ 1 -open and V is τ 2 -open. Let us consider m, n ∈ Z be arbi- Now, we recall some biological terms and try to use some of the above results in the growth process of an embryo from zygote until birth.

Definition 3.4.[19]
Mitosis is the process whereby one cell divides giving rise to two daughter cells each with 46 chromosomes. Definition 3.5. [19] Gastrulation is the process of forming the three primary germ layers from the epiblast involving movement of cells through the primitive streak to form endoderm and mesoderm.
Here; φ = Z = U 0 = V 0 , U m = X is the brain together with central nervous system of the whole organism and V n = Y is the other body parts of the whole organism except the brain and the central nervous system. Also, U 1 , U 2 ,... represent different development stages of the brain and the central nervous system; and V 1 , V 2 ,... represent different development stages of the other body parts except the brain and the central nervous system. Here, t 0 is the time of fertilization and T is the time of birth. It is important to note that before gastrulation, we have U i = V i . At the end; X and Y together form the whole organism, the baby R, i.e. X ∪ Y = R. Figure 1. represents it and it is due to Acharjee et al. [1]. Figure 1. is the modified version of the given figure by Nada and Zohny [2].
Acharjee et al. [3] showed that (R, h) is a bitopological dynamical system, where the map h : R → R (it is the map that represents mitosis process) is defined by Let R * be the postgastrulation part of the whole organism R, τ * 1 and τ * 2 are the relative topologies on R * and h * is the restriction of the map h on R * i.e. h * = h| R * . Then according to [3], (R * , τ * 1 , τ * 2 ) is a subspace of (R, τ 1 , τ 2 ) and the mapping h * : is a pairwise open map. In this section, we mainly focus on the bitopological dynamical system Again, the set X * and Y * are disjoint; since X * consists of brain and central nervous system cells whereas Y * consists of cells from other body parts except the brain and central nervous system. Thus, R * is expressed as the union of two non-empty disjoint sets X * and Y * . Hence, by theorem 2.1, the bitopological space (R * , τ * 1 , τ * 2 ) is pairwise disconnected. Now, let U * ,V * ⊂ R * be two developing stages of the embryo just after gastrulation such that U * ∈ τ * 1 and V * ∈ τ * 2 . After a certain time, U * will develop into X * and V * will develop into Y * . Let us consider a, b ∈ N be such that (h * ) a (U * ) = X * and (h * ) b (V * ) = Y * . Also, the set X * and Y * are disjoint due to the above paragraph. Thus, there exists a, b ∈ N, U * ∈ τ * 1 and

ON CONJECTURE 2 OF NADA AND ZOHNY
In conjecture 2 of [2], Nada and Zohny conjectured that a medical treatment should be started to stop cognitive anomalies at any step of growth based on the properties of local topological subspaces for the dynamical topology. In this section, we disprove their conjecture by using some new notions of Acharjee et al. [3] with medical evidences.
First we recall some biological terms.   According to [23], narrow definitions of MR restrict it to cases of non-progressive cognitive impairment detectable early after birth. Schaefer and Bodensteiner [24] found that a significant group of patients with MR have readily observable malformations of the brain. These malformations have in common the disruption of primary organizational processes in brain formation that occur very early in embryogenesis. Two of the most common anomalies listed by Schaefer and Bodensteiner [24] are Holoprosencephaly and Lissencephaly.
Definition 4.4. [19] Holoprosencephaly is a developmental defect where so much midline tissue for the face and brain has been lost that the two lateral ventricles fuse together and appear as one.
Definition 4.5. [25] Classical lissencephaly (previously type I) or generalized agyria-pachygyria is a severe brain malformation manifested by a smooth cerebral surface, abnormally thick cortex with four abnormal layers, widespread neuronal heterotopia, enlarged ventricles, and often agenesis or malformation of the corpus callosum.
Recently, Roessler et al. [16] found that mutations in the human Sonic Hedgehog gene cause Holoprosencephaly. Also, Lo Nigro et al. [15] and Reiner et al. [14] showed that classical or type I lissencephaly results from sporadic mutations in the LIS1 gene.
According to Oxford Dictionary of English [26], anomaly is something that deviates from what is standard, normal or expected. Thus, cognitive anomaly is synonymous with cognitive impairment.
Let x be a brain cell or central nervous system cell. Suppose, during the cell cycle of the cell x; a gene is mutated. The mitosis of this cell will produce two daughter cells y and z which are defected (here, by a defected cell we mean a cell that contains atleast one mutated gene). Thus, in this way a defected cell will lead to a group of defected cells ultimately leading to cognitive anomalies [14][15][16]. Acharjee et al. [3] showed that the bitopological space (R * , τ * 1 , τ * 2 ) is pairwise Hausdorff and therefore there is no structural relationship between brain cells together with central nervous system cells and cells from other body parts. Thus, to cure cognitive anomalies, medical treatments should be started on the brain cells or on the central nervous system cells. In this case, medical treatments on body cells will not cure cognitive anomalies as there is no structural relationship between brain cells together with central nervous system cells and cells from other body parts. Similar argument follows for defected body cells.
Thus, to cure cognitive anomaly, medical treatments should be done in the local bitopological subspace (S * , τ * 1 | S * , τ * 2 | S * ), where S * = U ∪V ⊂ R * , U is a developing stage of brain and central nervous system and V is a developing stage of other body parts without brain and central nervous system and the treatments must be particularly done in U. In this case, there may not be any role of the developing stage of V . Thus, we disprove conjecture 2 of Nada and Zohny [2], in the sense that to cure cognitive anomalies; medical treatments should be done on local bitopological subspace; not on local topological subspace as conjectured by Nada and Zohny [2].
Although, in the previous paragraph we claimed that to cure cognitive anomalies; medical treatments should be started on the brain cells or on the central nervous system cells, yet in some recent papers [30,31] researchers have developed methods that can convert adult skin cells directly to brain cells by reprogramming the cells in postnatal stages. These cells functionally integrate into the local circuit of brain cells and thus these methods become promising areas of treating brain diseases. But it is interesting to note that to convert a skin cell to a brain cell, the cell is needed to be reprogrammed. These experiments and methods partly verify our theories and claims. Since, biological science is a rapidly growing area, thus we do not nullify the possibilities of uses of reprogrammed cells as brain cells in prenatal stages to treat cognitive anomalies. Mostly, cognitive anomalies can be treated by the treatments of brain cells or central nervous system cells. We hope that our theoretical claims may be found to be true in future since Optogenetics [27], a new brainstorming research area in neuroscience, uses lights on neurons to study various brain activities for Parkinson's disease.

OPEN QUESTION
In this section, we state a question which is open to the readers of this paper.
Q. Is there any example that pairwise connectedness of a bitopological space (X, τ 1 , τ 2 ) does not imply forward iterated connectedness or backward iterated connectedness or pairwise iterated connectedness in a bitopological dynamical system (X, f ) ?

CONCLUSION
In this paper, we introduced the concepts of forward iterated connectedness, backward iterated connectedness and pairwise iterated connectedness in a bitopological dynamical system and we established some relationships with pairwise connectedness. Also, we proved that the developing stage after gastrulation is forward iterated disconnected in the development of a human embryo from the zygote. Later, we disproved the conjecture 2 of Nada and Zohny [2] by using the concept of pairwise Hausdorffness together with medical evidences of gene mutation. According to Adolphs [28], the cure of psychiatric and neurological diseases is one of the unsolved problems of neuroscience. In this case, in near future our theories may become the backbone of treatments for developmental anomalies of human brain. To make this a reality, we need deep research in the area of human embryo development process through the process of mitosis and bitopological dynamical system.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.