CHAPTER ONE

1.0 INTRODUCTION

Ideas that are now classified as topological were expressed as early as 1736. Towards the end of 19th century, a distinct discipline developed, which were referred to in Latin as geometria situs (geometry of place) or analysis situs (Greek-Latin for ‘’picking apart of place). The later acquired the modern name of topology. By the middle of 20th century, topology has become an important area of study within mathematics. Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the19th century. Cantor in addition to setting down the basic ideas of set theory considers point sets in Euclidean space as part of his study of Fourier series. Maurice Rene Frechet, unifying the work on function spaces of Cantor, Volterra, Arzela, Hadamard, Ascoli and others introduced the metric space in his doctoral dissertation in 1906. A metric space is now considered a special case of a general topological space. In 1914, Felix Hausdorff coined the term ‘’topological space’’ and gave the definition of what is now called Hausdorff space. In current usage, a topological space is slight generalization of Hausdorff space, given in 1922 by Kazimierz Kuratowski.

The Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space. However, metric space as a space of mathematical objects has more specific definition of distance and topological space has the definition of nearness.

After some experience with the real and complex analysis it becomes apparent that the development of the theory of metric depends to a great extent simply on the notion of distance between the numbers in the system of real numbers. The real numbers as a system have three basic properties that are of great importance which can be categorized into three: the algebraic (addition, subtraction, multiplication e.t.c), the order property (notion of distance) and the completeness property (concept of supremum). All these properties are well entailed in a set of axioms which completely characterized the real number system. Assume R is a set of real numbers, P is a set of positive numbers and the function ‘+’ and ‘ . ‘that are addition and multiplication respectively defined on

R× R→R which satisfy the properties of real numbers.

The Field Axiom (algebraic properties). ∀ x,y,z∈R, we have that x+y=y+x ∈R (Cummutative law of addition)

(x+y)+z=x+(y+z)∈ R (Associative law of addition)

∃ 0∈R such that x+0=x ∈ R (Identity law of addition)

∃ w∈ R such that x+w=0 (Inverse law of addition)

x.y=y.x∈R (Cummutative law of multiplication)

(x.y)z=x(y.z)∈ R (Associative law of multiplication)

∃ 1 such that 1≠0 and x.1=x (Identity law of multiplication)

∃ w∈ R such that x.w=1 (Inverse law of multiplication)

x.(y+z)=x.y+x.z∈R (Distributive law)

Any set that satisfy the above set of axioms under the functions is called a field.

The Order Axiom (Notion of Distance). Let P be any subset of R i.e P⊂R, then the following holds, ∀ x,y,z ∈P

(x,y)⇒x+y∈P⊂R

(x,y)⇒x.y∈P⊂ R

(x)⇒-x∉P

(x∈R) ⇒(x=0) or (x∈P) or (-x∈R)

Any subset P satisfying the field and order axioms is called an ordered field

The Completeness Axiom (Upper Bound Axiom). It is essential to have a good understanding of some concepts before stating this last axiom; we say that b is an upper bound of R if ∀ x∈ R, we say x≤b. A number c is called the least upper bound for R if it is an upper bound for R if c≤b for each upper bound b of R. From this, it is clear that upper bound of R is unique if it exist. The completeness axiom states that every non empty set R which has an upper bound has a least upper bound. This axiom is a single axiom, and it is this axiom that distinguishes the real number from other ordered field and with this, it is called a complete ordered field.

From the discussion of properties of real numbers, it follows that the order property is the beginning of a metric space and more attention will be paid to this axiom in this project as it deals with the notion of distance which is among the main basis of this research.

This project is divided into four chapters. Chapter one deals with the general introduction including aim, objectives, significance, limitations and some basic definitions and examples. In chapter two, some relevant literatures were reviewed as it concerns the main topic of discussion. In chapter three we formulated some relationship between topological space and metric space using the map structure of Nigeria as an example. In chapter four we discussed the implication of the formulated relationship as established in chapter three and with respect to the theoretical properties of the two algebraic spaces. Lastly we provided summary and conclusion of the research including recommendation for further researches.

1.1 AIM OF THIS RESEARCH.

This research work is aimed at:

Understanding the basis of the two spaces in question and to identify and construct some metric and topological spaces using the Nigeria map.

1.2 OBJECTIVES OF THIS RESEARCH.

The objectives of this research are:

To construct metric and topological spaces using the Nigeria map

To identify some metric and topological properties using the Nigeria map

To establish some relationship between geo political zones using some topological concepts.

1.3 SIGNIFICANCE OF THIS RESEARCH.

The structure of Nigeria map is divided into geo political zones, states and local government areas. The need to identify the geographical distance (approximate distance in kilometers travelling by car on main roads) between states and local government areas is important for both developmental purposes and for location of various projects as well as social services. This project provides some analysis for both the locations and sizes of centers with respect to the six geo political zones of Nigeria.

1.4 SCOPE AND LIMITATION OF THIS RESEARCH.

There are many abstract spaces in the study of mathematics, but for the purpose of this research, metric spaces and topological spaces are the main focused spaces of this research. The research is also limited to study geographical distances (approximate distance in kilometers travelling by car on main roads) of places (state capitals) considered on the Nigeria map.

1.5 SOME BASIC DEFINITIONS AND EXAMPLES.

1.5.1 Metric Space.

A metric space (X,λ) is a non empty set X of real numbers with a real valued function λ defined on the set X such that ∀ x,y,z∈X

λ(x,y) ≥0

λ(x,y)=0 iff x=y

λ(x,y)=λ(y,x)

λ(x,y)≤ λ(x,z)+λ(y,z)

Condition 3 is called the symmetry property and condition 4 is called triangular inequality which asserts that the shortest route from a place x to a place y is the direct route.

Example: The real line R denote the set of real numbers and let d:R× R→R be defined by d(x,y)= |x-y| ∀ x,y,z∈ R then d is a metric on R and is called the usual metric R .

Example: Let X be an arbitrary non empty set define θ: X×X →R by

θ = {█(1 if x≠y@0 if x=y)┤

∀ x,y in X then θ is a metric on X called the discrete metric on the set X.

Example: Let R^2 be the Euclidean plane and let λ: R^2×R^2→R defined by λ(x,y)= { (〖x_1-y_1)〗^2+〖〖(x〗_2 〖-y〗_2) 〗^2 }^(1⁄2) =[∑_(i=1)^2▒(x_i-y_i )^2 ]^(1⁄2)

Where x=(x_1,x_(2))) and y=(y_1 〖,y〗_2) are arbitrary elements on R^2 called the Euclidean metric.

1.5.2 Open Ball

Given a metric space (X,d) and any x_o∈X .Take radius r>0 (r∈ R) then B_r(x_o) =B(r,x_o) ={y ∈ R: d(x_o,y ) 0 (r∈R) then B_r(x_o) = B(r,x_o) = {y ∈ R: d(x_o,y ) ≤r} is a closed ball with radius r, centered at x_o.

Example: B_1(0) = {u= (x,y )∈ R^2:〖 x〗^2+y^2≤1} with the given metric d(x,y ) = [∑_(i=1)^2▒(x_i-y_i )^2 ]^(1⁄2) where x=(x_1,x_2) and y=(y_1 〖,y〗_2).

By B_1(0) = {u= (x,y )∈ R^2:〖 x〗^2+y^2≤1} it implies that the unit disc centered at the origin together with the boundary of the disc.

1.5.4 Sphere.

Given a metric space (X,d) and any x_o∈ X. Take radius r >0 (r∈ R) then S_r(x_o) =S(r,x_o)={y ∈ R: d(x_o,y )=r} is a sphere with radius r, centered at〖 x〗_o.

Example: {u= (x,y )∈ R^2:〖 x〗^2+y^2=1} which implies the unit circle centered at the origin.

1.5.5 Open Set

Let E⊆X then E is an open set in X with respect to the given metric and ∀ x∈E there exist r>0 (r ∈ R) such that B_r (x)⊆E.

Example: Let X=R and A=[0,1 ┤) and R is endowed with the metric defined by θ ={█(1 if x≠y@0 if x=y)┤ with this metric, A=[0,1 ┤) is open in R

Solution: Let x_o∈A be arbitrary. We must find an open ball B_r(x_o) with center at〖 x〗_o and radius r>0 such that B_r(x_o) ⊂A. We choose r=□(1/2) (or 0<r<1) and so r=□(1/2)0 such that〖 B〗_r (a)⊆A. Then A^0=int(A) is the set of all interior points of A.

Example: Let (R,λ) be the real line with the usual metric. Let M=(-4,0├]∪┤[10,15)⊂ R be a subset of R. Estimate

The interior of M.

The closure of M.

The limits points of M.

The boundary point of M.

Solution: M^o(interior of M). From the content of M, all x∈M are interior point of M except 0 and 10 because an open ball with the usual metric is B_r (x)=(x-r,x+r) .Considering 0 ⇒B_r (0)=(0-r,0+r), even if 0-r∈M but 0+r∉M. So B_r (0) is not totally contained in M. Similarly considering 10 ⇒B_r (10) = (10-r, 10+r), where 0+-r ∈ M but 0-r ∉ M. Therefore M^o= (-4, 0)∪(10, 15)

1.5.7 Limit Points

Let A⊆X then ∀x,∈X [B_r (x)∖{x} ]⋂A≠ϕ is called limit points of A or cluster point ofA. Then the set of all limit points of A is called the derived set ofA.

Example: From the set M, all x∈M are limit points but -4,15∉M because an open ball centered at -4 and 15 implies -4+r, 15-r ∈M even if -4,15 ∉M. So the set of limit points M^I=[-4,0]∪[10,15].

1.5.8 Closed Set

Let F⊆X, then F is closed in X with respect to the given metric if F contains all its limit points

1.5.9 Closure of Set

Let A⊆X then closure of A={x∈X:x∈A or x is a limit point of A} i.e A∪A^I is the closure ofA.

Example: we compute the closure from the example above. Closure of M=M∪M^I=(-4,0├]∪┤[10,15)∪ [-4,0]∪[10,15].

1.5.10 Boundary Point

Let ⊆X ,p∈X is a boundary point of A if given r>0, for all r∈R, B_r (p)∩A = ϕ and B_r (p)∩A^c ≠ ϕ and ∂(A) is called the boundary of A

1.5.11 Neighborhood

Let (X,λ) be a metric space, and let x∈X. A subset N_xis said to be a neighborhood of x if x∈ N_x and if there exist an open ball B_r (x), for some r>0, centered at x such that B_r (x)⊂N_x.

1.5.12 Topology

A collection τ of subsets of non empty set X is called a topology for X if the following axioms are satisfied

X ∈ τ and ϕ∈ τ

∩t_i ∈ τ where t_i are subsets of τ

∪t_i ∈ τ

1.5.13 Topological Space

A topological space (X ,τ) is the combination of a non empty set X and the collection τ, of subsets of X i.e τ= {O_α:α∈ Δ and O_α⊆ X} such that

ϕ , X ∈ τ

⋂_(αi i=1 )^n▒O_αi ∈ τ for each O_αi ∈ τ

⋃_(α∈ Δ )▒O_α ∈ τ whenever O_α ∈ τ

Example: X ≠ ϕ , τ_O={ϕ,X } then (X, τ_O) is trivial topological space and τ_O is the weakest topology on X.

Example: X ≠ ϕ, τ_1, P=(X) is the power set of X i.e the set of all possible subsets of X then (X,τ_1 ) is discrete topological space and is the strongest topology on X.

1.5.14 Limit

Let {x_n } be a sequence of points in a metric space (X,λ). A point x^* ∈ X is called a limit of the sequence if for r>0 there exist a positive integer N such that x_n ∈B_r (x^* ) for all n≥ N where B_r (x^* )={x∈X:λ(x^*,x)0 there exist an integer N>0 such that for all m,n>0 we have, λ(x_m,x_n)0 there exist r>0 such that λ_y((f(x_o), f(x)) <ε whenever λ_x (x_o,x)0, there exist N_o such that d(x_n,x_m) N_o then d(x_n,x_(m_o ))0(a∈R,a≠0),|a|=|-a | (a∈R) and|a+b|≤|a|+|b| (a,b∈R). From the absolute value properties, the geometric interpretation of |x-y| is the distance from x to y and if we define the distance function ρ by ρ(x,y)=|x-y| (x,y∈R) then the properties of a absolute value function have the following consequences for any point x,y,z ∈R

ρ(x,x)=0 (That is the distance from a point to itself is 0)

ρ(x,y)>0 (x≠y) (The distance between two distinct points is strictly positive)

ρ(x,y)=ρ(y,x) (The distance from x to y is equal to distance from y to x )

ρ(x,y)≤ρ(x,z)+ ρ(z,y) (Triangle inequality)

This inequality says that going from x to y directly never takes longer than going x to z and z to y. He added that ρ is called a metric and defined metric space as follows: Let M be any set, a metric for M is a function ρ with domain M×M and range contained in [0,∞)┤ such that

ρ(x,x)=0 (x∈M)

ρ(x,y)>0(x,y∈M,x≠y)

ρ(x,y)=ρ(y,x) (x,y∈M)

ρ(x,y)≤ρ(x,z)+ρ(z,y) (x,y∈M)

If ρ is a metric for M, the ordered pair is called a metric space.

Rudin (1987) in his own contribution to metric spaces opined that the most familiar topological spaces are metric spaces and he defined it as a set X in which a distance function (or metric) ρ is defined, with the following properties:

0≤ρ(x,y)0 and ∂(x,y)=0 if and only if x=y.(positivity)

∂(x,y)=∂(y,x). (symmetry)

∂(x,y)+∂(y,z)≥∂(x,z). (triangle inequaliity)

A primary challenge of the approach is that established biological models of similarity do not form metrics. Most biological similarity functions require similar features with greater positive numbers. Metric require the distance of more similar objects to be closer to zero. The similarity models are often derived from probabilistic methods that difficult to algebraically transform into metrics.

Clarkson (2005) used metric space for nearest neighborhood in building data structure. The problem of nearest neighborhood search is to build a data structure for a set of objects so that, given a query object q, the nearest object in the set to q can be found quickly.

That is, suppose U is a set and D is a distance measure on U, a function that takes pair of elements of U and returns a non negative real number. Then given a set S⊂D of size n, the nearest neighborhood searching problem is to build a data structure so that, for an input query point q∈U, an element a∈S is found with D(q,a)≤D(q,x)forall x∈S. We will call the members ofS sites, to distinguish them from other members of U, and say that the answer a is nearest in S to q. If put in another way, we define D(x,S) as min{D(x,s)|s∈S}, then, we seek the site S such that D(q,s)=D(q,S).He therefore define metric space as the distance from D of (U,D) satisfying the following conditions, for all x,y,z∈U

Non negativity: D(x,y)≥0;

Small self distance:D(x,x)=0;

Isolation: x≠y implies D(x,y)>0;

Symmetry: D(x,y)=D(y,x);

The triangle inequality:D(x,z)≤D(x,y)+D(y,z).

Many instances of nearest neighborhood searching have an associated metric space. However, it is worth noting that if any of the conditions fails while others holds, there is a natural associated function like the metric. If condition 3(isolation) fails; here (U,D) is called a pseudometric space, partition U into equivalence classes base on D, where x and y are equivalent if and only if D(x,y)=0 with the natural distance D([x],[y] )=D(x,y) on the classes, the result is a metric space. If condition 4 (symmetry) also fails, (U,D) is a quasi metric space. The related measure D(x,y)=(D(x,y)+D(y,x))∕2 will satisfy symmetry, and so yields a metric space also.

Over time, the idea of metric spaces has not vary as it concern the notion of distance irrespective of the field it is been used. In all works of researchers reviewed in this project, it is obvious that non negativity, symmetry and triangle inequality are the highlights of metric space. Although, individual’s mode of presentation may differ in some sense but they have the same underlying meaning.

CHAPTER THREE

3.0 METRIC SPACES AS TOPOLOGICAL SPACES

The treatment of a metric space as topological space is so consistent that it is almost a part of the definition. About any point x in a metric space M we define the open ball of radius (r>0) about x as the set B(x,r)={y∈M d(x,y)<r}. These open balls generate a topology of M because the open balls are the building blocks of metric space topology making it a topological space. The fundamental structure on a topological space is not distance but nearness. The metric topology on M is the coarsest topology on M relative to which the metric is a continuous map from the product of M to the non negative real numbers. A topological space which can arise in this way is called metrizable. Since metric spaces are topological spaces, one has a notion of continuous function between metric spaces and can also be directly defined using limits of sequences.

The metric spaces have three properties: topological properties, which are preserved under homeomorphism; uniform properties, which are preserved under uniform homeomorphism and the metric properties which are preserved under isometries. Metric properties are by nature restricted to the category of metric spaces. Any property that can be define solely in terms of open set is a topological property and these properties usually extend to the to the category of topological space and the properties and notions (topological) include continuity, convergence, closure e.t.c. The uniform properties such as uniform continuity, uniform convergence, equicontinuity, total boundedness and completeness fall between the topological and metric properties.

3.1 METRIZABILITY OF TOPOLOGICAL SPACE.

A metric space (X,λ) can give rise to a topological space (X,τ) where τ is defined to be the collection of all subsets which are open in the sense of open and close set. τ is the topology on X called metric topology induced by λ. Suppose we go the other way by starting with a metric space, we are given a topological space (X,τ), then we need check if there exist a metric λ which can be define on X such that the topology induced by the λ is τ. So if this is true, then the space X is said to be metrizable. Otherwise, X is non metrizable.

It is now a task to know when a topological space is metrizable. A very interesting theorem which helps in detecting it is called the metrization theorem. One of the most important metrization theorems is the famous Urysohn’s metrization theorem which clearly gives the criteria suffices for metrizability.

3.2 URYSOHN’S METRIZATION THEOREM.

Let X be a normal topological space which is second countable. Then X is metrizable or it can also be stated that a space X is second countable if it has a countable base

However not all metrizable spaces are second countable. Also the Uryhsohn’s lemma also serves as back up for theorem. The lemma states that let A and B be disjoint closed subsets of a normal space X, then there is a continuous real valued function f:X→[0,1] such that f(a)=0 for all a∈A and f(b)=1 for all b∈B. This idea of metrizability by the Urysohn’s metrization theorem is to construct a homeomorphism f between the given topological space X, which is normal and second countable, and a subspace of the Hilbert cube, which is metrizable. Then f(x) will be metrizable as a subspace of a metrizable space and so is X since homeomorphic to a metrizable space.

3.3 METHOD OF CONSTRUCTION

3.3.1 STRUCTURE OF NIGERIA MAP AS A METRIC SPACE.

A number of interesting metrics can be defined on a set X; a metric emphasizes some features of interest while ignoring others. For instance, let the set N be the structural map of Nigeria and the state capitals including FCT are points or elements of the set Nigeria. Three possible metrics on our set are d_g, which measure geographical distance (road); d_c, which measure travel cost; and d_t, which measure travel time t. For the purpose of this construction, d_g(measure of distance by road which imply other two metrics) is the metric define on N such that (N,d_g ) forms a metric space.

Set N is the Nigeria structural map. The 36 state capitals including FCT are points in N i.e

Umuahia=s_1 Ado Ekiti=s_13 Lafia=s_25

Yola=s_2 Enugu=s_14 Minna=s_26

Uyo=s_3` Gombe=s_15 Abeokuta=s_27

Awka=s_4 Owerri=s_16 Akure=s_28

Bauchi=s_5 Dutse=s_17 Oshogbo=s_29

Yenogoa=s_6 Kaduna=s_18 Ibadan=s_30

Makurdi=s_7 Kano=s_19 Jos=s_31

Maiduguri=s_8 Katsina=s_20 Port Harcourt=s_32

Calabar=s_9 Birnin Kebbi=s_21 Sokoto=s_33

Asaba=s_10 Lokoja=s_22 Jalingo=s_34

Abakaliki=s_11 Ilorin=s_23 Damaturu=s_35

Benin City=s_12 Ikeja=s_24 Gusau=s_36

Abuja=s_37

Then N ={s_1,s_2,s_3,……………,s_37 } and the metric space (N,d_g )=|s_α+s_β | for all α,β,γ that takes value in [1,37]. So (N,d_g ) is a metric space if for all s_α,s_(β,) s_γ ∈N if

d_g (s_α,s_β)≥0

d_g (s_α,s_β )=0 if and only if α=β

d_g (s_α,s_β )=d_g (〖s_β,s〗_α)

d_g (s_α,s_β )≤d_g (s_α,s_γ )+d_g (s_γ,s_β). We need to verify the statements.

VERIFICATION I

d_g (s_α,s_β )=|s_α-s_β |≥0 , for all α,β,γ∈ [1,37]⊂R .

|α|=|β|=|γ|≥0. This implies that the distance from one state capital to another in Nigeria will never be negative.

ILLUSTRATION: Let α=37,β=24 and d_g be the road distance which is the metric, then the metric space d_g (s_α,s_β ) is define as

d_g (s_37,s_24 )=|s_37-s_24 |=|Abuja-Ikeja| .Meaning that the distance between Abuja and Ikeja.

From the distance chart, d_g (s_37,s_24 )=|Abuja-Ikeja|=879km≥0

VERIFICATION II

d_g (s_α,s_β )=|s_α-s_β |=0 if and only if α=β

But 〖 d〗_g (s_α,s_α )=|s_α-s_α |=|0|=0.Suppose d_g (s_α,s_β )=0

Then |s_α-s_β |=0 ⟹ s_α-s_β=0 and so s_α=s_(β ).Obviously, the distance of a particular point to itself is 0.

ILLUSTRATION: If α=β=33 , this implies that s_33=Sokoto and so 〖 d〗_g (s_α,s_β )=〖 d〗_g (s_33,s_33 )=0, then |s_33-s_33 |=|0|=0. This illustrates that the distance from a particular point in Sokoto to the same point is absolutely 0

VERIFICATION III

d_g (s_α,s_β )=d_g (s_β,s_α ).

d_g (s_α,s_β )=|s_α-s_β |=|-(s_α-s_β ) |=|-s_α+s_β |=|s_β-s_α |

ILLUSTRATION: Let α=20,β=30 and so d_g (s_α,s_β ) is given by

d_g (s_20,s_30 )=|s_20-s_30 |=|Katsina-Ibadan|=1052km=|Ibadan-Katsina|

by the distance chart. This illustration shows that the distance from Katsina to Ibadan is still the same from Ibadan to Katsina.

VERIFICATION IV

d_g (s_α,s_β )≤d_g (s_α,s_γ )+d_g (s_γ,s_β).

d_g (s_α,s_β )=|s_α-s_β |=|s_α-s_γ+s_γ-s_β |.

≤〖|s〗_α-s_(γ|)+|s_γ-s_β |

And so d_g (s_α,s_β ) ≤ d_g (s_α,s_γ )+d_g (s_γ,s_β ) . This shows that the shortest distance between two points is the direct route.

ILLUSRATION: Let α=33,β=8,γ=18 which correspond to Sokoto, Maiduguri and Kaduna respectively. Then d_g (s_α,s_β )≤d_g (s_α,s_γ )+d_g (s_γ,s_β). So we have d_g (s_33,s_8 )≤d_g (s_33,s_18 )+d_g (s_18,s_8 )

|s_33-s_8 |≤|s_33-s_18 |+|s_18-s_8 |

|Sokoto-Maiduguri|≤|Sokoto-Kaduna|+|Kaduna-Maiduguri| . From the distance chart |1204km|≤|487km|+|876km|

1204km≤1363km

This illustration implies that the shortest distance Sokoto to Maiduguri is the direct route, not through any third point.

Hence, the verification I to IV prove that the structure of Nigeria map with respect to 〖 d〗_g (distance by road) as the metric define on the set N (Nigeria) is indeed a metric space.

3.3.2 OPEN BALLS IN THE MAP STRUCTURE OF NIGERIA.

Nigeria is divided into six geo political zones namely:

North West (G_z1) = {Dutse 〖(S〗_17), Kaduna (S_18 ), Kano (S_19 ), Katsina 〖(S〗_20), Birnin Kebbi 〖(S〗_21), Sokoto 〖( S〗_33), Gusau 〖(S〗_36)}.

North East (G_z2) = {Yola (S_2 ), Bauchi (S_5 ), Maiduguri (S_8 ), Gombe (S_15 ), Jalingo 〖(S〗_34), Damaturu 〖(S〗_35)}.

North Central (G_z3) = {Makurdi 〖(S〗_7), Jos (S_17 ), Lokoja 〖(S〗_22), Ilorin (S_23 ), Lafia 〖(S〗_25), Minna 〖(S〗_26), Abuja (S_37)}.

South West (G_z4) = {Ado Ekiti (S_13), Ikeja 〖(S〗_24), Abeokuta (S_27), Akure (S_28), Osogbo (S_29), Ibadan 〖(S〗_30)}.

South East (G_z5) = {Umuahia 〖(S〗_1), Awka (S_4), Abakaliki (S_11 ), Enugu (S_14 ), Owerri 〖(S〗_16)}

South South (G_z6) = {Uyo (S_3 ), Yenogoa (S_6 ), Calabar (S_9 ), Asaba 〖(S〗_10), Benin City 〖(S〗_12), Port Harcourt (S_32)}.

These geo political zones are the subsets of the set Nigeria and subspaces of the metric space(N,d_g), that is, these subsets are also metric spaces in their own respect when the metric d_g is define on them. Recall that an open ball B_r={y∈X:d(x_o,y)0. we follow this idea into each of the geo political zones.

OPEN BALL IN NORTH WEST (G_z1)

Let (G_z1,d_g ) be a metric space (subspace of metric space(N〖,d〗_g )). Given a point S_36 (Gusau) ∈ G_z1 and a positive real number r (r>0), then the set B_r (S_36 )={S_r∈G_z1:d_g (S_36,S_r )<r} is an open ball in〖 G〗_z1.

PROOF: If we choose an arbitrary r to be 600km

The set B_600km (S_36 )={S_r:d_g (S_36,S_r )0), then the set B_r (S_15 )={S_r∈G_z2:d_g (S_15,S_r )<r} is an open ball in〖 G〗_z2.

PROOF: If we choose an arbitrary r to be 500km

The set B_500km (S_15 )={S_r:d_g (S_15,S_r )0), then the set B_r (S_26 )={S_r∈G_z3:d_g (S_26,S_r )<r} is an open ball in〖 G〗_z3.

PROOF: If we choose an arbitrary r to be 500km,

The set B_500km (S_26 )={S_r:d_g (S_26,S_r )0), then the set B_r (S_30 )={S_r∈G_z4:d_g (S_30,S_r )<r} is an open ball in〖 G〗_z4.

PROOF: If we choose an arbitrary r to be 300km,

The set B_300km (S_30 )={S_r:d_g (S_30,S_r )0), then the set B_r (S_1 )={S_r∈G_z5:d_g (S_1,S_r )<r} is an open ball in〖 G〗_z5.

PROOF: If we choose an arbitrary r to be 200km,

The set B_200km (S_1 )={S_r:d_g (S_1,S_r )0), then the set B_r (S_6 )={S_r∈G_z6:d_g (S_6,S_r )<r} is an open ball in〖 G〗_z6.

PROOF: If we choose an arbitrary r to be 400km,

The set B_400km (S_6 )={S_r:d_g (S_6,S_r )0) such that B_r (x)⊂A.

Clearly, the six geo political zones form open sets in the set N (Nigeria) considering the metric space (N〖,d〗_g ) since each geo political zone is a subset of Nigeria and there exist open ball in each of the subsets (geo political zones).

3.3.4 THEOREM: Let (X,d) be a metric space. Then

An arbitrary union of open sets in X is an open set in X.

A finite intersection of open sets in X is an open set in X.

PROOF I. Let {B_α:α∈∆} (where ∆ is any index set) be a collection of open sets in X. Let B=⋃_(α∈∆)▒B_α . We want to show that B is open in X.

This implies that we have to show that for any arbitrary point x∈B, there exist some r>0 such that x∈B_r (x)⊂B. So, x∈B be arbitrary. But then, x∈B = ⋃_(α∈∆)▒〖B_α⟹x∈B_(α_o ) 〗 for some α_o∈∆. By hypothesis, B_(α_o ) is open. So, it follows that there exist some r>0 such that x∈B_r (x)⊂B_(α_o )⊂⋃_(α∈∆)▒B_α . Hence B=⋃_(α∈∆)▒B_α is an open set in X.

IMPLICATION

From the first part of the theorem, the union of all geo political zones (open sets) forms the entire country (Nigeria) which is also an open set because any point (state capital) that we take in the union of all geo political zones (Nigeria) is a member of one of the geo political zones.

PROOF II. Let {B_i:i=1,2,….,n} be a collection of n open sets in X and let B=⋂_(i=1)^n▒〖B_i.〗 We want to proof that B is an open set in X. This means that for an arbitrary x∈B, we can find some r>0 such that x∈B_r (x)⊂B. So let x∈B=⋂_(i=1)^n▒〖B_i 〗be arbitrary. This implies x∈B_i for each i. But each B_i is an open set in X. So, there exists some r_i>0 such that x∈B_ri (x)⊂B_i for each i=1,2,….,n. We have that x∈B_r (x)⊂⋂_(i=1)^n▒〖B_i 〗=B. So, B= ⋂_(i=1)^n▒〖B_i 〗is an open set in X.

IMPLICATION

Obviously, the six geo political zones are disjoint subsets of the set Nigeria and so the intersection of these subsets (geo political zones) is an empty set ϕ (no one state capital is in two or more geo political zones in Nigeria). Thus the intersection of these open sets is empty set ϕ which is also open in the set N (Nigeria).

3.5.2 THEOREM: Let (X,d) be a metric space. A subset A of X is an open set in X if and only if A is a union of open balls.

FORWARD PROOF (⟹)

Let A be an open set in (X,d). We want to show that A is a union of open balls. Let x∈A, since A is an open set in X, there exist some real number r>0 such that B_r (x)⊂A. In the same manner, it follows that for each y∈A we can find r_y>0 (r, a real number depending on the particular y) such that B_(r_y ) (y)⊂A then clearly A=⋃_(y∈A)▒{y} ⊂⋃_(y∈A)▒B_(r_y ) (y)⊂A, because 〖y∈B〗_(r_y ) (y) and each B_(r_y ) (y)⊂A. So, A⊂⋃_(y∈A)▒B_(r_y ) (y)⊂A, and this implies A=⋃_(y∈A)▒B_(r_y ) (y) is the union of open balls.

BACKWARD PROOF (⟸)

Let A be expressible as the union of open balls in X, that is, A=⋃_(x∈A)▒〖B_(r_x ) (x)⊂X.〗 We want to prove that A is an open set in X. We have to show that for arbitrary x∈A we can find some r_x>0 such that B_(r_x ) (x)⊂A. So, let x∈A be arbitrary, since A=⋃_(y∈A)▒〖B_(r_y ) (y),〗 x must belong to at least one of the open balls B_(r_y )∈A. Without loss of generality, we may assume that B_(r_x ) (x)⊂B_(r_o ) (y)⊂⋃_(y∈A)▒B_(r_y ) (y)=A. Hence, A is open.

IMPLICATION

The metric space (N,d_g ) has six open balls with respect to the six geo political zones. So we can arrive some open sets G_p1={B_r 〖(S〗_36)∪B_r 〖(S〗_15)∪B_r 〖(S〗_26)} and G_p2={B_r 0)∪B_r 〖(S〗_1∪B_r 〖(S〗_6)}, the union of all open balls in the upper layer of the map of Nigeria (northern part G_p1) and the lower layer of the map of Nigeria (southern part G_p2). Without loss of generality G_p1 and G_p2 are subsets of N (Nigeria) and subspaces of the metric space (N,d_g ).

It is clear that the northern part G_p1 and southern part G_p2 together with empty set ϕ and the universal set N are open. Thus, the collection of all these open sets forms a topology in N (Nigeria) and is given by τ={G_p1,G_p2,ϕ,N}. From this; the required topological space is not far fetch. The combination of the set N (Nigeria) and it’s topology τ is a topological space (N,τ) if :

ϕ∈τ,N∈τ

⋃_(i=1)^2▒〖G_pi∈τ 〗

⋂_(i=1)^2▒〖G_pi∈τ.〗

Indeed, the structural map of Nigeria is a topological space since the union of the northern and southern part of the entire country (Nigeria) is in the collection τ. Also the intersection of the northern and southern part is an empty set ϕ which is also in the collection τ. The map structure of Nigeria as a topological space is induced by the metric d_(g )of the metric space (N,d_g ) where the open balls and open sets serve as medium between these two abstract spaces and the topological space (N,τ) is metrizable.

CHAPTER FOUR

4.0 SUMMARY

In this project work, we see that the open balls and open set are medium in metric space topology, that is, a linear arrangement exists from metric space to open ball, open set and then to topological space.

This project work has focused on the Nigerian Map having capital cities as points in the set N as a metric space and topological space with respect to road distance as the distance function d_g.

In the map of Nigeria as a space containing open balls, there are six open balls such that each open ball is from one geopolitical zone. These balls vary in radius which automatically shows that the sizes of the balls also vary, these variations in the size of the balls have some implications that has to be analyzed (radius and centres of balls).

4.1 VARIATION IN RADIUS OF OPEN BALLS

In the map structure of Nigeria, six open balls were constructed with different radius. If these balls are categorized into two; Northern open balls, (open balls in North West (G_z1), North Central (G_z2) and North East (G_z3) ) and Southern open balls (open balls in South West (G_z4 ), South East (G_z5) and South South (G_z6)). By comparing the categories of the open balls (Northern open balls and Southern open balls), we see that the smallest radius of northern open ball is more than the biggest radius in the Southern open balls which imply that there are more land mass in Northern Nigeria than the Southern part of Nigeria.

This variation of radius and size of open balls also account for the variation of development in Northern and Southern part of Nigeria. Generally, the Southern part of Nigeria is more developed than the Northern part of Nigeria because there is this concentration of development because of less land mass but that of the North is scattered. Also, there is good road Network (link) between cities in the south than the Northern part of Nigeria.

The availability of land in the North gives room for this part of the country to be more developed than the South in years to come if the necessary social amenities are put in place or if all that has made the South developed are also made available in the Northern part of Nigeria.

4.2 CENTRES OF OPEN BALLS

The choice of centres of the open balls is strictly on geometric idea. Each centre of the balls is a capital city that is almost at the centre of other capital cities in each geopolitical zone.

4.3 RECOMMENDATIONS

This project work recommends that more resources should be expended to bridge the gap between the Northern and Southern part in terms of physical structure due to the huge difference in land mass.

Also on the idea of centres of open balls in each geopolitical zone, if the government intends to position a capital project (may be a central work station) in each geopolitical zone, this project recommends the centres of the open balls of each geopolitical zones for easy access to the central work station. For instance, if the government decides to build a regional library in each geopolitical zone, for the purpose of easy access for all from their capital cities, the centres of open balls will be the best location to site the libraries.

4.3 CONCLUSION

Metric space topology is an effective mathematical structure in analyzing any arbitrary space particularly a geographical space which this project work has done with respect to road distance from capital cities. This project has built the idea for further researches on metric space topology and geographical spaces with respect to other parameters such as are embodied in some theoretic properties of the topology. This includes investigation on compactness, connectedness, homeomorphism, sepratability e.t.c and their relevance to physical structures such as maps.

APPENDIX I

APPENDIX

II

RESEARCH PAPERS EXTRACTED FROM THE PROJECT

1. Metrization process as an optimization tool: a case of Nigerian map and distribution of distances from state capitals by Kazeem A.O. and Ibrahim A.A. (Submitted for publication as at November 2011)

2. Some topological properties of Nigerian map: a case of metrization using state capitals by Kazeem A.O. and Ibrahim A.A. (Submitted for publication as at November 2011)

3. Application of metric topology to the study of geographical distance in Nigerian map by Kazeem A.O. and Ibrahim A.A . (Submitted for publication as at November 2011)

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