الجـامعــــــــــة اﻹســـــﻼميــة بغــزة The Islamic University of Gaza
عمادة البحث العلمي والدراسات العليا Deanship of Research and Graduate Studies
كـليــــــــــــــــــــة العـلـــــــــــــــــــــوم Faculty of Science Master of Mathematical Science ماجستيـــــــر العلـــــــوم الـريـاضـيــة
Topological Properties of the Cantor Spaces
الخصائص التبولوجية لفضاءات كانتور
By
Ahamed M. Alnajjar
Supervised by
Prof. Dr. Hisham B. Mahdi Professor of Mathematices
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Sciences.
October/2018
إقــــــــــــــرار
أنا الموقع أدناه مقدم الرسالة التي تحمل العنوان:
Topological Properties of the Cantor Spaces ﺍلﺨﺼائص ﺍلتبولوﺟﻴة لﻔضاءات ﻛانتوﺭ
أقر بأن ما اشتملت عليو ىذه الرسالة إنما ىه نتاج جيدي الخاص، باستثناء ما تمت اإلشارة إليو حيثما ورد،
وأن ىذه الرسالة ككل أو أي جزء منيا لم يقدم من قبل االخرين لنيل درجة أو لقب علمي أو بحثي لدى أي
مؤسدة تعليمية أو بحثية أخرى.
Declaration
I understand the nature of plagiarism, and I am aware of the University’s policy on this.
The work provided in this thesis, unless otherwise referenced, is the researcher's own work, and has not been submitted by others elsewhere for any other degree or qualification.
اسم الطالب: أحمد دمحم خالد النجار Student's name: Ahamed M. Alnajjar
التهقيع: :Signature
التاريخ: :Date
i
Abstract
ﻓﻲ هذه الرسالة ، نتحدث بشكل رئيسﻲ عن ﻓضاءات كانتور ومكعب كانتور ﻷية مجموعات غير منتهيه ﻓﻲ X ﻓان مكعب كانتور تم تعريفه على انه مجموعات الطاقة من (P(X من X. وهذا الفضاء له تمثيﻼت أخرى ويتكون اﻷساس لهذا الفضاء من كل الفترات التﻲ تكون على شكل [ك, م] حيث ان ك منتهيه و م غير منتهيه وحيث تم استخدام هذا اﻷساس بفعاليه ودرسنا خصائص الفضاءات الكانتور و مكعب كانتور وأنواع مختلفه من مكعب كانتور ,وأثبتنا أن الفضاء كانتور هو الفضاء الوحيد (ما يصل إلى التماثل) التﻲ تجمع بين خصائص الفضاء المنفصل ، والكمال ، والمدمج ، والمتر المترابط تما ًما وأثبتنا أن µ تكون تبولوجﻲ ألكساندر وف ﻓإنها تكون مغلقه ﻓﻲ 푿 .
وعﻼوة على ذلك ﻓإن أصغر تبولوجﻲ ألكساندر وف هﻲµ Abstract
In this thesis, we mainly talk about the Cantor space and the Cantor cube. For any infinite set X, the Cantor cube is defined as the power set P(X) of X with the product topology. This space has other representation 2X . The base for this space consists of all intervals of the form [K,F ] where K is finite set and F is cofinite set, and this base has been used effectively. We study the properties of the Cantor space, the Cantor cube 2X and the properties of different types of cubes. We prove that the Cantor space is the only space (up to homeomorphism) that combines the properties of totally disconnected, perfect, compact, and metric space. We prove that τ on X is Alexandroff topology iff τ is closed in 2X . Moreover, if τ is topology on X, then the closure τ of τ in 2X is the smallest Alexandroff topology in X containing τ .
iii Acknowledgements
After God, I appreciate the greatest appreciation of my parents who continued my support and pray for me to continue until the end. I thank my wife and all my brothers, friends and loved ones for sharing my scientific joy. I owe them my deepest love. My thanks go to professor Hisham Mahdi who supported me and supported me to continue this thesis. And always gave me suggestions and guidance, and I would like to thank all the faculty in the Department of Mathematics, especially professor Asaad Asaad, professors Ayman al-Sakka, Dr. Ahmed Mabhouh and profesor, God’s mercy upon him, Eissa al-Habil.
iv Contents
Abstract iii
Acknowledgements iv
Contentsv
Introduction1
1 Preliminaries5
1.1 Properties of topological spaces...... 5
1.2 Directed sets and nets...... 10
1.3 Product topology...... 12
2 Alexandroff Spaces 15
2.1 Partially ordered sets...... 15
2.2 Alexandroff topology and minimal open neighborhoods...... 18
2.3 T0-Alexandroff space viewed as poset...... 21 2.4 Product of Alexandroff space...... 24
v 3 Cantor Space 27
3.1 Construction of the Cantor set...... 28
3.2 Equivalence representation of the Cantor space...... 29
3.2.1 The sequence representation...... 29
3.2.2 The representation of C as power set of N ...... 31 3.3 Properties of the Cantor space...... 31
4 The Cantor Cube of Infinite set X 39
4.1 The Cantor cube 2X ...... 39
4.2 Properties of different types of cube...... 42
4.3 More on the Cantor cube 2X ...... 44
Bibliography 53
vi Introduction
The Cantor set is a subset of the interval [0,1] and has many definitions and many different constructions. The Cantor set is a set of points in the line. It was defined by the German mathematician Georg Cantor in 1874 and introduced in 1883. The
Cantor set has laid a modern foundation in the category of the topology. Although
Cantor originally provided a purely abstract definition, the most accessible is the middle-thirds or ternary set construction. For the Cantor space, we attempt to a systematic study of analytic topologies over the natural numbers N (or any count- able set X). We identify every subset of N with its characteristic function. This implies that the power set P(N) is identified as the Cantor space 2N. Now and for more general, Let X be an infinite set, P(X), the power set of X is defined as the product space 2X = {0, 1}X , the collection of all function from X to {0, 1} where
{0, 1} has discrete topology. In this case, the space 2X is called a cantor cube of
X. The idea of this thesis is how to systematically study the types of topology within the Cantor cube. The thesis depends basically on the results of the paper
”Alexandroff topologies viewed as closed sets in a Cantor cube”. The goal is to develop this research on other characteristics of the topology. Alexandroff spaces were first introduced by P. Alexandroff in 1937 under the name of Diskrete Raume
1 (discrete space). An Alexandroff topology is a topology has the additional property arbitrary intersection of open sets is open. In this paper we present some results of
Alexandroff topologies. This thesis consists of four chapters. Chapter one contains three sections about preliminaries that will be used in the remainder of this the- sis. In section one, we talk about the basic definitions and theorems of topological spaces. In the second section, we give an introduction to directed sets and nets.
We study some important theories that will be used in this thesis. The last section take about the product topology. Chapter two studies the Alexandroff topology and its topological properties. This chapter has been divided into four sections.
The first section studies partially order sets. In the second section, we study the
Alexandroff topology and minimal open neighborhoods. It is easy to verify that a T topology is Alexandroff Topology iff for every x ∈ X, the set Nx = {U : x ∈ V and V ∈ τ} is an open set. Nx is called the irreducible (or minimal) neighborhood of x and one can easy check that a collection A of subsets of X has minimal set T Bx of x ∈ X in A iff {V ∈ A : x ∈ U} ∈ A. In third section we prove that each T0-Alexandroff space has a poset in 1 − 1 and onto way such that each one is completely determined by the other. In last section, we study the product of
Alexandroff topology. Chapter three is talking about the Cantor set. In the first section, we study the construction of Cantor set. And we will show that it is non empty set. Thus, the Cantor set is a subset of I formed from deleting accountable union of open interval. If we consider C as a subspace of I with its usual topology,
C is called the Cantor space. We will study the properties of this space in the next section. But in this section, we give some representation of a space equivalent
2 (up to homeomorphic) to the Cantor space. From now an, we will use any rep- resentation of them as the Cantor space. In second section, we study equivalence representation of the Cantor space. There are many equivalence representations but we will give only two of them. One representation is the sequence represen- tation and second one is the representation of C as a power set of N. In the last section, we study the properties of the Cantor space. The Cantor space contains some important properties. The Cantor set is a closed subset in [0, 1], totally dis- connected and a perfect space. Chapter four consists of three sections. The first one is talking about the Cantor cube 2X . For every K, F ⊆ X, we define the interval [K,F ] = {A ∈ P(X): K ⊆ A ⊆ F }. The collection of all intervals [K,F ] where K is finite and F is cofinite is a basis for the product topology on 2X . These intervals are in fact clopen in 2X . In the second section, we study the properties of different types of cube. We study the Cantor space 2N, the Cantor cube 2X and
A the cube I . We prove that these cubes are compact, Hausdorff, and T4-spaces. The Cantor cube 2X are second countable, separable and metrizable, and the cube
IA where A is countable are second countable, separable and metrizable. In last section, we give more details of the Cantor cube 2X . The starting point of this work is the fact that Alexandroff topologies on X exactly to closed set in 2X . The collec- tion of clopen topologies is particularly simple, since they essentially correspond to topologies over finite sets (both results were known for the case of a countable set
X). Moreover, we prove that if τ is a topology on X, then the closure τ of τ in 2X is also a topology on X. Thus the closure τ of τ in 2X is the smallest Alexandroff topology on X containing τ.
3 Chapter 1
Preliminaries Chapter 1
Preliminaries
In this chapter, it is necessary to define the following concepts and facts in topol- ogy such as, open set, closed set, cluster point, continuous function, characteristic function and product topology.
1.1 Properties of topological spaces
Definition 1.1.1. (Willard, 1970) Let X be a nonempty set, and τ a collection of subsets of X. Then τ is a topology on X if the following three conditions hold:
1. ∅ and X are in τ
2. A finite intersection of elements of τ is in τ.
3. Arbitrary union of elements of τ is in τ.
The subsets in τ are called open sets, and the complement of open sets are called closed sets. The pair (X, τ) (or simply X, if there is no confusion) is called a
Definition 1.1.2. (Willard, 1970) Let A be a subset of X. Then the interior of
A in X (denoted as A˚ or int(A)) is the union of all open subsets of A. That is int(A) = S{B : B is open and B ⊆ A }.
Definition 1.1.3. (Willard, 1970) Let A be a subset of X. Then
T 1. The closure clX (A) or (A) of A is defined as clX (A) = {G : G is closed and G ⊇ A}.
T 2. The frontier (or boundary) F rx(A) of A is defines F rx(A) = clx(A) clx(X − A).
3. A ⊂ XA is called dense if A = X.
Definition 1.1.4. (Willard, 1970) The space X is called separable if there exists a countable dense set A˚ in X.
Definition 1.1.5. (Willard, 1970) Let S be a subset of a topological space (X, τ). T Let τS = {S B : B ∈ τ}. τS is topology on S and (S, τS) is called a subspace of (X, τ).
Definition 1.1.6. (Willard, 1970) Let U be a subset of X and x ∈ int(U), then we say that U is a neighborhood (briefly, nhood) of x. The collection Ux= {U ⊆ X : U is neighborhood of x} is called the neighborhood system of x. A collection Bx ⊆ Ux is called neighborhood base at x if for every U in Ux, there exists a neighborhood
B ∈ Bx such that B ⊆ U.
6 Definition 1.1.7. (Willard, 1970) The discrete topology on X is the topology in which all sets are open. The trivial topology on X is the topology on X in which
∅ and X are the only open sets.
Definition 1.1.8. (Willard, 1970) Let B be a subset of a topological space X.A point x in X is a cluster point or an accumulation point of B if every neighbourhood of x contains at least one point of B different from x itself.
Definition 1.1.9. (Willard, 1970) A point x ∈ X is said to be a cluster point (or accumulation point) of a sequence (xn)n∈N if for every neighbourhood V of x and every n ∈ N, there is some n0 ∈ N such that xn0 ∈ V .
Definition 1.1.10. (Willard, 1970) Let X and Y be topological spaces. A function f : X −→ Y is called continuous at x if for each open set V in Y containing f(x), there exists an open set U in X containing x such that f(U) ⊆ V .
Definition 1.1.11. (Willard, 1970) Let X and Y be topological spaces. Then f : X −→ Y is called open (resp. closed) map if for each open (resp. closed) set A in X, f(A) is open (resp. closed) set in Y .
Definition 1.1.12. (Willard, 1970) Let X and Y be topological spaces. A function f : X −→ Y is called homeomorphism if and only if f is continuous, open, one to one, and onto map. We say X and Y are homeomorphic and (denote as X ∼ Y ) if there exists a homeomorphism between X and Y .
Remark 1.1.13. (Willard, 1970) If f : X −→ Y is a homeomorphism, then f −1 :
Y −→ X is also a homeomorphism.
7 Definition 1.1.14. (Willard, 1970) Let A be a subset of a nonempty set X.A function χA(x):X −→ {0, 1} defined as 1, if x ∈ A; χA(x) = 0, if x∈ / A. is called a characteristic function of A.
Definition 1.1.15. (Willard, 1970) A topological space X is said to be a To-space if for any two distinct points x, y ∈ X with x 6= y exists open sets U in X such that
U containing x and not y or there exists open set V in X such that V containing y and not x.
Definition 1.1.16. (Willard, 1970) A topological space X is said to be T1-space if for any two distinct points x, y ∈ X with x 6= y there exist two open sets U and
V such that U containing x and not y and V containing y and not x.
Definition 1.1.17. (Willard, 1970) A topological space X is said to be Hausdorff or T2-spacee if for any two distinct points x, y ∈ X with x 6= y, there exists two open sets U containing x and V containing y such that U T V = ∅.
Lemma 1.1.18. (Willard, 1970) Let A be a subset of a topological space X. Then the characteristic function of A is continuous if and only if A is both open and closed where the topology on the set {0,1} is the discrete.
Proof. (⇒) Suppose that χA is continuous. Since {1} is open and closed in {0, 1},
−1 then χA {1} = A is both open and closed set in X.
8 (⇐) Suppose that A is both open and closed subsets of X. The open sets in {0, 1}
−1 −1 −1 are ∅, {0, 1}, {0}, {1}. Clearly χA ({∅}) = ∅, χA ({0, 1}) = X, χA ({1}) = A,
−1 −1 χA ({0}) = X −A, and all of them are open sets in X. So χA (x) is continuous.
Definition 1.1.19. (Willard, 1970) A topological space X is said to be discon- nected if there exists non-empty open sets U and V of X such that U T V = ∅ and X = U S V . We say {U, V } is a disconnection of X. A topological space X is called connected if it is not disconnection, i.e. if there are no disconnection of X.
If A ⊆ X we say A is connected iff (A, τA) is connected where τA is the relative topology on A. Explicitly, A is disconnected in (X, τ) iff there exists U, V ∈ τ such that U T A = ∅, U T A = ∅,A T U T V = ∅ and A ⊂ U S V
Lemma 1.1.20. (Willard, 1970) If a finite space is T1, then it is discrete.
Proof. Suppose that X is finite and T1−space. Then ∀x ∈ X, {x} is closed set [ in X. Pick a ∈ X. Then X − {a} is finite subset of X and X − {a} = {x} x6=a is a finite union of closed sets, so it is closed. Hence {a} = (X − {a}){ is open in X. Since a ∈ X is arbitrary, the singleton sets are open in X and hence X is discrete.
Definition 1.1.21. (Willard, 1970) A set A in a space X is perfect in X if A is closed and dense in itself (:= each point of A is an accumulation point of A).
Definition 1.1.22. (Willard, 1970) Let X be a topological space. A cover of X is [ a collection ξ = {U i}i∈I such that X = Ui. A subcover of ξ, is a sub collection i=1 σ ⊆ ξ which is a cover of X.
9 Definition 1.1.23. (Willard, 1970) Let (X, τ) be a topological space. Then X is compact if every open cover has a finite sub caver.
Theorem 1.1.24. (Willard, 1970) Let (X, τ), (Y, τ1) be two topological spaces, and let f : X −→ Y be continuous function. If B ⊆ X is compact, thenf(B) is compact in Y .
Theorem 1.1.25. (Willard, 1970) Let (X, τ) be Hausdorff space.
1. If B ⊆ X is compact, then B is closed.
2. If B is compact then a subset K ⊆ B is compact if and only if K is closed in
B.
Corollary 1.1.26. (Willard, 1970) Let X, Y be topological spaces. If X is compact,
Y Hausdorff space, and f : X −→ Y is a continuous and bijective, then f is homeomorphism. That is f is open.
Proof. Let U be on open subset of X, then X\U is closed in X, and by Theorem
1.1.25, it is compact. By Theorem 1.1.24, then f(X\U) is compact, and since Y is Hausdorff, f(X\U) is closed in Y . But f(X\U) = Y \f(U). Therefor Y \f(U) is closed and f(U) is open in Y .
1.2 Directed sets and nets
Definition 1.2.1. (Willard, 1970) A set D with a relation ≤ is called a directed set if it satisfies the following:
1. (Reflexive) ν ≤ ν for any ν ∈ D.
10 2. (Transitive) If ν1 ≤ ν2 and ν2 ≤ ν3, then ν1 ≤ ν3.
3. (Direct) for any ν1, ν2 ∈ D, there exists ν3 such that ν1 ≤ ν3 and ν2 ≤ ν3.
Definition 1.2.2. (Willard, 1970) Let X be a nonempty set. A net in X is a map
ν : D −→ X from a directed set D into X. We write, (νλ)λ∈D ⊆ X. If X equipped with a topology, a net (νλ)λ∈D converges to x ∈ X and we write ν −→ x if for any nhood U of x, there exists λ0 ∈ D such that ∀λ ≥ λ0, νλ ∈ U.A λ0−tail of a net is the set {xλ : λ ≥ λ0}. So ν −→ x if any nhood U of x contains some tail of the net.
Definition 1.2.3. (Willard, 1970) Let A be a subset of topological space and
(νλ)λ∈D a net in X. We say ν is eventually in A iff A contains some tail of ν. A net
ν is called frequently in A iff for every λ ∈ D there exists λ0 ≥ λ such that νλ ∈ A. Equivalently λ−tail intersects A ∀λ ∈ D.
Lemma 1.2.4. (Willard, 1970) Let F be a subset of a topological space X. Then F is closed if and only if whenever (νλ)λ∈D is a net in F converges to x, then x ∈ F .
Proof. (⇒) Suppose that F is closed, and suppose to contrary that ∃ a net (νλ)λ∈D in F such that νλ −→ x and x∈ / F . Then x ∈ X − F which is an open nhood of x. Hence ∃λ0 ∈ D such that ∀λ ≥ λ, xλ ∈ X − F therefor xλ ∈/ F ∀λ ≥ λ0 which is a contradiction.
⇐ Suppose that F is not closed in X. Then ∃x ∈ F − F . Hence ∀ nhood U of X, T T U F 6= ∅. Pick xu ∈ U F . Then we get (xu)u∈Ux be in F where Ux = {u : u is nhood x}. If we defined ≤ on Ux as U ≤ V if V ⊆ U, then Ux is a directed and hence (xu)u∈D is a net in F . Moreover, xu −→ x where x∈ / F .
11 Theorem 1.2.5. (Willard, 1970) Let X be a topological space and A ⊆ X. Then x ∈ A if and only if there is a net (νλ)λ∈D in A such that νλ −→ x.
Theorem 1.2.6. (Willard, 1970) Let X and Y be topological spaces. Then f :
X → Y is continuous if and only if for any net (νλ)λ∈D in X such that νλ −→ x, we get f(νλ) −→ f(x) in Y .
1.3 Product topology
Definition 1.3.1. (Willard, 1970) Let {Xα|α ∈ ∆} be a collection of nonempty sets. We defined the cartesian product set as:
Y [ Xα = {x : ∆ −→ Xα|x(α) = xα ∈ Xα, ∀α ∈ ∆}. α∈∆ α∈∆
th For any α ∈ ∆, xα is called the α -coordinate of x. For any β ∈ ∆ The map
Y th πβ : Xα −→ Xβ defined by πβ(x) = xβ is called the β - projection map. α∈∆
−1 T −1 Remark 1.3.2. (Willard, 1970) If Uβ ⊆ Xβ and Vγ ⊆ Xγ then πβ (U) πγ (V ) = Y (Q X ) × U × V = {x ∈ X : x ∈ U and x ∈ V }. α6=βα6=γ α β γ α β β γ γ α∈∆
Definition 1.3.3. (Willard, 1970) Let {(Xα, τα)|α ∈ ∆} be collection of topologi- Q cal spaces and X = α Xα be the cartesian product set. The collection B consists of all subsets of X in the from π−1(U ) T π−1(U ) T ··· T π−1(U ) = (Q X )× α1 α1 α2 α2 αn αn α6=αi α
Uα1 × · · · × Uαn where Uαi is open in Xαi from a base for a topology an X called the product topology and denoted by τp. From now on, if {(Xα, τα): α ∈ ∆} be a Y collection of topological spaces, then Xα will be the product space. α
12 Theorem 1.3.4. (Willard, 1970) Let {(Xα, τα)|α ∈ ∆} be a family of topological spaces. Then πβ is continuous and open for all β ∈ ∆.
−1 Y Proof. for any Vβ ∈ τβ. Then πβ (Vβ) = Uα such that Uα = Xα for all α 6= β α∈∆ −1 and Uβ = Vβ ∈ τβ. So πβ (Vβ) ∈ τp. Hence πβ is continuous. In addition, Let
V ∈ τp. Then πβ(V ) = Xβ or πβ(V ) = Uβ, Uβ ∈ τβ. In both cases, the image of Q open set in Xα is open in Xβ, so πβ is open.
Theorem 1.3.5. (Willard, 1970) Let {Xα : α ∈ ∆} be a family of topological Q spaces, then α∈∆ Xα is Hausdorff iff Xα is Hausdorff ∀α ∈ ∆.
Q Proof. Let x = (Xα)α∈∆, y = (Yα)α∈∆ be two distinct element in α∈∆ Xα. There exists a β ∈ ∆ such that xβ 6= yβ. Since Xβ is a Hausdorff space, there exist open T −1 subsets U, V ∈ Xβ such that xβ ∈ U, yβ ∈ V and U V = ∅ then x ∈ πβ (U) −1 −1 T −1 −1 T −1 and y ∈ πβ (V ),πβ (U) πβ (V ) = πβ (U V ) = πβ (∅) = ∅. Since by Theorem
−1 −1 1.3.4 πβ is continuous function πβ (U) and πβ (V ) are open sets which separate x Q and y. Therefor α∈∆ Xα is Hausdorff
Theorem 1.3.6. (Willard, 1970) Let {Xα : α ∈ ∆} be a family of topological Y spaces, then Xα is compact iff Xα is compact ∀α ∈ ∆. α∈∆
Theorem 1.3.7. (Willard, 1970) Let {Xα : α ∈ ∆} be an arbitrary collection of Y connected topological spaces. Then the topological product Xα is connected. α∈∆
13 Chapter 2
Alexandroff Spaces Chapter 2
Alexandroff Spaces
2.1 Partially ordered sets
Definition 2.1.1. (Davey and Priestley, 1990) Let P be a non-empty set and ≤ be a relation on P . We say ≤ is a partial order (or simply order) on P if ≤ satisfies the following:
1. (Reflexive)if x ≤ x for all x ∈ P .
2. (Antisymmetric) if x ≤ y and y ≤ x then x = y for all x, y ∈ P .
3. (Transitive) if x ≤ y and y ≤ z then x ≤ z for all x, y, z ∈ P .
The pair (P, ≤) is a partially ordered set or (or simply a poset).
Definition 2.1.2. (Davey and Priestley, 1990) Let a, b ∈ P , the element b is called a maximal if whenever b ≤ c then b = c, and the element a is called a minimal if whenever c ≤ a, then c = a. The set of all maximal (resp. minimal) element in P is denoted by M(P ) or simply M (resp. m(P ) or simply m).
15 Definition 2.1.3. (Davey and Priestley, 1990) Let A ⊆ P , then (A, ≤|A) is a partial order subset of P and the set of maximal elements in A (resp. the set of minimal elements in A) is denoted by M(A) (resp. m(A)). Two elements x, y in
P are comparable x ≤ y or y ≤ x. Other wise they are incomparable.
Definition 2.1.4. (Davey and Priestley, 1990) A subset A of a poset P is called convex if ∀x, y ∈ A and z ∈ X such that x ≤ z ≤ y we have z ∈ A. We say x is cavered by y and we write x −→ y if x < y and x ≤ z ≤ y whenever x < z ≤ y, then z = y.(x < y means x ≤ y and x 6= z).
Definition 2.1.5. (Davey and Priestley, 1990) In a poset X, if there exists an element > ∈ P such that x ≤ > for all x ∈ P , then > is called a maximum (or top) element. If there exists an element ⊥ ∈ P such that ⊥ ≤ x for all x ∈ P , then
⊥ is called a minimum (or bottom) element.
Definition 2.1.6. (Davey and Priestley, 1990) A subset G of a poset P is a down set (or a lower set) if whenever x ∈ G and y ≤ x, then we have y ∈ G. A subset
W of a poset P is an up set (or a upper set) if whenever x ∈ W and x ≤ y, then we have y ∈ W . We define the down set ↓ x = {y ∈ P : y ≤ x}, and the up set
↑ x = {y ∈ P : x ≤ y}.
Example 2.1.7. (Davey and Priestley, 1990) The set N of all natural numbers forms a poset under the usual order ≤ on R. Similarly, the set of integers Z, rationales Q and real numbers R under the usual order ≤ form posets.
Representation of a poset: We can present the poset by diagram, represent each element by a small circle (or a
16 dot). Any two comparable elements are joined by lines in such a way that if a ≤ b, then a lies below b in the diagram. Non-comparable elements are not joined. Thus, there will not be any horizontal lines in the diagram of a poset.
Example 2.1.8. The set {2,5,6,10} under the division relation forms a poset with a diagram as below:
Figure 2.1: (P, ≤)
Definition 2.1.9. A poset P satisfies the ascending chain condition (ACC) if for any increasing sequence x1 ≤ x2 ≤ · · · ≤ xn ≤ · · · in P , there exists k ∈ N such that xk = xk + 1 = ··· . It satisfies the descending chain condition (DCC), if for any decreasing sequence x1 ≥ x2 ≥ · · · there exist, k such that xk = xk+1 = ··· . If a poset satisfies both ACC and DCC, we say P is of finite chain condition (FCC).
1 1 Example 2.1.10. (Isaacs, 2009) Let P1 = { n |n ∈ N} and P2 = {1 − n |n ∈ N} whit usual order. Then both P1 and P2 are linearly ordered poset. Moreover P1 satisfies The ACC but not the DCC and P2 satisfies the DCC but not the ACC.
17 Example 2.1.11. Each finite poset is of FCC.
2.2 Alexandroff topology and minimal open neigh-
borhoods
Firstly, if N is a nhood of x (N ∈ Ux) such that N ⊆ U for all U ∈ Ux, then N must be open and called minimal open nhood of x. Let (X, τ) be a topological space. In general, an arbitrary intersection of open sets need not be open and equivalently, an arbitrary union of closed sets need not be closed. Moreover, if x ∈ X, then it is not n that x has a minimal nhood.
Definition 2.2.1. (Alexandroff, 1937) Let X be a topological space. If X satisfies the property that any intersection of open sets is open, then X is called Alexandroff space.
Theorem 2.2.2. (Speer, 2007) X is Alexandroff iff ∀x ∈ X, there exists a minimal open nhood Nx such that Nx ⊆ U ∀U ∈ τ, with x ∈ U.
Proof. (⇒) Let Bx = {U ∈ τ : x ∈ U}. Then Bx is a collection of open sets. Since \ X is Alexandroff, then the set Nx = B is open set containing x. Moreover,
B∈Bx Nx ⊆ B ∀B ∈ Bx. ⇐ Let B be a collection of open sets and B = T B. If B = ∅, we done. Otherwise, let x ∈ B, then ∃Nx open set containing x and Nx ⊆ U ∀x ∈ U ∈ τ. If V ∈ B, then x ∈ V and V is open. Hence Nx ⊆ V . Since V is arbitrary in B, then Nx ⊆ V T ∀V ∈ B and Nx ⊆ B = B. Hence B is open set. Thus X is Alexandroff.
18 Example 2.2.3. 1. Let (X, τ) be the discrete topological space, then X is Alexan-
droff space.
2. Any finite space is Alexandroff space.
Theorem 2.2.4. (Arenas, 1999) Let X be Alexandroff space. Then X is T1 iff X has the discrete topology .
Proof. (⇒) Let X be a T1 Alexandroff space, and x ∈ X, then for all x ∈ X, {x} is closed and hence X − {x} is open. Let a ∈ X be arbitrary, then ∀x 6= a, X − {x} \ \ is open. So X − {x} is open since X is Alexandroff. But X − {x} = {a}. x6=a x6=a Thus, ∀a ∈ X, {a} is open in X and hence X is discrete.
(⇐) Any discrete space is T1.
Definition 2.2.5. (Speer, 2007) Let X be a nonempty set, x ∈ X and A a collec- tion of subsets of X. We say Bx ∈ A is a minimal set in A for x if Bx ⊆ U ∀U ∈ A T and x ∈ U. In this case, Bx = {U ∈ A : x ∈ U}.
Remark 2.2.6. (Speer, 2007) One can easy check that a collection A of subsets of T X has minimal set Bx of x ∈ X in A iff {U ∈ A : x ∈ U} ∈ A.
Example 2.2.7. Let X = {1, 2, 3, 4} and A = {{1}, {1, 2}, {1, 3}, {3, 4}}.
Since T{U ∈ A : 1 ∈ U} = {1} T{1, 2} T{1, 3} = {1} ∈ A, a minimal set
B1 of 1 in A exists. Similarly, B2 = {1, 2},B4 = {3, 4}. For x = 3, note that T{U ∈ A : 3 ∈ U} = {1, 3} T{3, 4} = {3} ∈/ A.
Theorem 2.2.8. (Speer, 2007) Let A be a collection of subsets of X. If for each x in X there exists a minimal set Bx ∈ A of x, then A is a basis for a topology on X and this topology is Alexandroff topology.
19 T Proof. Let U, V ∈ A and x ∈ U V . Let Bx be a minimal set in A containing x, T [ [ and so Bx ⊆ U and Bx ⊆ V . Hence, Bx ⊆ U V . Moreover, X = Bx ⊆ A. x∈X A∈A So A is a base for a topology on X. Let Bx be open and hence minimal open. Therefor by Theorem 2.2.2, X is Alexandroff.
Corollary 2.2.9. (Speer, 2007) Let (X, τ) be an Alexandroff space and for x ∈ X, let Nx be the minimal open nhood of x. Then the collection B = {Nx : x ∈ X} is base for the Alexandroff topology τ.
Proof. By Theorem 2.2.8, B is a base for a topologyτ ´ on X. Since B ⊆ τ, then
τ´ ⊆ τ. On other hand, if U ∈ τ and x ∈ U then x ∈ Nx ⊆ U (since τ is Alexandroff). Then U ∈ τ´, then τ ⊆ τ´.
√ Example 2.2.10. Let X = R2. For any x = (a, b) ∈ X, |x| = a2 + b2. Define ∀r ≥ 0, B(0, r) = {x ∈ X : |x| < r} and B(0, r) = {x ∈ X : |x| ≤ r}. Not that ∀x ∈ X, x ∈ B(0, |x|). Moreover if x ∈ B(0, |y|) for some y ∈ X = R2, then |x| ≤ |y|. So if z ∈ B(0, |x|), then |z| ≤ |x| ≤ |y| then z ∈ B(0, |y|) then
B(0, |x|) ⊆ B(0, |y|). This implies that the collection A = {(0, |x|): x ∈ X} has a minimal set ∀x ∈ X which is Bx = B(0, |x|). Therefore by Theorem 2.2.8 A is a base for an Alexandroff topology on R2.
n p 2 2 2 Remark 2.2.11. (Speer, 2007) In more general we can take X = R , |x| = a1 + a2 + ··· + an and B(0, r) = {y ∈ Rn : |y| ≤ r} for r ≥ 0. Then A = {B(0, |x|): x ∈ Rn} is a base for some Alexandroff topology on Rn.
Recall that a space is connected if the only sets which are clopen are the empty set and the whole space.
20 Theorem 2.2.12. (Arenas, 1999) Let X be an Alexandroff space and ∀x ∈ X, let
NX be the minimal open nhood of x. Then NX is compact and connected subset of X.
Proof. Let {Uα}α∈A be an open caver of Nx. Then x ∈ Uα0 for some α0 ∈ A. So
Nx ⊆ Uα0 .(X is Alexandroff). Therefor {Uα0} is subcover of Nx from {Uα}α∈A consist of one element. So Nx is compact. S T S If Nx = U V where U, V ∈ τNx , and U V = ∅, then x ∈ Nx = U V , then T ´ ´ ´ T ´ x ∈ U or x ∈ V . If x ∈ U = Nx U, U ∈ τ, then Nx ⊆ U, U = Nx U = Nx, therefor V = ∅. So Nx is connected.
2.3 T0-Alexandroff space viewed as poset
Theorem 2.3.1. (Alexandroff, 1937) Let (X, ≤) be a poset and let B = {↑ x : x ∈
X}. Then B is a base for a topology on X and this topology is a T0-Alexandroff topology denoted by τ(≤).
T Proof. For any x ∈ X, set Ux =↑ x = {y : x ≤ y}. Let x ∈ Uy Uz. Then x ≥ y T and x ≥ z. If t ∈ Ux, then t ≥ x and hence t ≥ y and t ≥ z. Thus t ∈ Uy Uz. T This implies that Ux ⊆ Uy Uz Thus {Ux} is a base for a topology on X. Now we show the topology is an Alexandroff. Let x ∈ X and suppose U ∈ τ is open containing x. Since B is a base, ↑ x ⊆ U. Hence ↑ x is smallest open nhood of x.
Theorem 2.3.2. (Alexandroff, 1937) Let (X, τ) be a T0-Alexandroff space for any a, b ∈ X, define a ≤ b iff a ∈ {b}. Then (X, ≤) is a poset and τ(≤) is the original
21 T .
Proof. since a ∈ a then a ≤ a. If a ≤ b and b ≤ a then a ∈ b and b ∈ a. Thus, a ⊆ b and b ⊆ a. Therefore a = b. Suppose to contrary that a 6= b. Then either
{ T { a∈ / Nb or b∈ / Na (X is T0). If a∈ / Nb, then b ∈ b = a ∈ Nb . Hence b ∈ Nb Nb which is a contradiction and hence a = b. Now, if a ≤ b and b ≤ c, then a ∈ b and b ∈ c. Thus a ∈ b ⊆ c = c. Thus a ≤ c. Therefore (X, ≤) is a poset.
Claim: Nx =↑ x . [To see this, if y ∈ Nx and U is any open set containing x, T then Nx ⊆ U and so U {y} = {y} 6= ∅. Thus x ∈ y and so x ≤ y. This implies that y ∈↑ x and Nx ⊆↑ x. On the other hand if z ∈↑ x, then x ≤ z and so x ∈ z. T T Since Nx is open set containing x, then Nx {z}= 6 ∅. Thus Nx {z} = {z} and so z ∈ Nx. Therefore ↑ x ⊆ Nx]. By this claim, we proved that τ(≤) is the original topology τ.
Theorem 2.3.3. If (X, τ(≤)) is a T0-Alexandroff. space with corresponding poset (X, ≤), then U is open (resp. F is closed ) in X in the category of topology iff U is up set (resp. F is down set) in the category of poset.
Remark 2.3.4. Clearly, from Theorem 2.3.1 and Theorem 2.3.2 every T0-Alexandroff space has Correspondence poset. So T0-Alexandroff space in the category of topol- ogy can be viewed in the category of poset as shown in the following example:
Example 2.3.5. 1. Let X = {a, b, c, d} with the partial order a ≤ b, a ≤ c and
d ≤ c as shown in Figure1 below:
22 Figure 2.2: (X, ≤)
B = {↑ a, ↑ b, ↑ c, ↑ d} = {{a, b, c}, {b}, {c}, {d, c}}
Then is abase for a T0- Alexandroff topology on X. The induced topology is
τ(≤) = {∅,X, {a, b, c}, {b}, {c}, {d, c}, {b, c, d}, {b, c}}.
The set A = {a, b, d} is down set which is not up, so it is closed and not open.
Note that X − A = {c} ∈ τ in (X, ≤). Clearly {c} is up set
2. Let X = {a, b, c, d}, with the T0- Alexandroff topology
τ = {∅,X, {a, b, c}, {b}, {c}, {b, d, c}, {b, c}}.
We can find the specialization order as follows: {a} = {a}, {d} = {d},
{b} = {b, a, d}, and {c} = {c, a, d}, so a ≤ b, d ≤ b, a ≤ c, and d ≤ c and
hence the figure of the poset X is
23 Figure 2.3: (X, ≤τ )
Theorem 2.3.6. (Mahdi, 2010) Let (X, τ(≤)) be a T0- Alexandroff space, x ∈ X and A ⊆ X. Then
1. For x ∈ X, {x} =↓ x.
2. A˚ = {x ∈ A :↑ x ⊆ A} = S{↑ x :↑ x ⊆ A}.
3. A = S{↓ x : x ∈ A}.
4. A´ = A\{x : x is maximal in A}.
2.4 Product of Alexandroff space
If X and Y are two topological spaces, then the product space X × Y is T0 if and only if both X and Y are T0. The following theorem constructs new Alexandroff spaces from old.
Theorem 2.4.1. (Arenas, 1999) Let (X, τ) and (Y, τ1) be topological spaces. If X and Y are Alexandroff space, then X × Y is Alexandroff space with minimal nhood
24 Nx×y = Nx × Ny.
n Y Corollary 2.4.2. If (Xi, τi) is Alexandroff space. For i = 1, 2, ··· , n, then Xi i=1 is Alexandroff space.
Proof. We can use Theorem 2.4.1 and Mathematical induction.
Remark 2.4.3. The Alexandroff property is finitely productive. So if {(Xα, τα): Y α ∈ ∆} is a collection of Alexandroff spaces, then Xα need not be Alexandroff α∈∆ space.
Theorem 2.4.4. (Mahdi, 2010) Let (X, τ1(≤1)) and (Y, τ2(≤2)) be two T0-Alexandroff spaces with corresponding posets (X, ≤1), (Y, ≤2) respectively. Then the specializa- tion order ≤p of the T0-Alexandroff space X × Y coincides with the Coordinatewise order induced by the product of the corresponding posets.
Proof. We will use the fact that in a T0-Alexandroff space, x ≤ y iff y ∈ Nx. Now
(a, b) ≤p (c, d) iff (c, d) ∈ Nx(a, b) = Nx(a) × Nx(b) iff c ∈ Nx(a) and d ∈ Nx(b)
25 Chapter 3
Cantor Space Chapter 3
Cantor Space
The basic Cantor set is a subset of the interval [0,1] and has many definitions and many different constructions. The Cantor set is a set of points in the line.
It was defined by the German mathematician Georg Cantor in 1874 and intro- duced in 1883, the Cantor set has laid a modern foundation in the category of the topology. We attempt to a systematic study in this chapter on construction of the Cantor set, representation of the Cantor space and properties of the Cantor space. Although Cantor originally provided a purely abstract definition, the most accessible is the middle-thirds or ternary set construction. Begin with the unit interval I0 = [0, 1]. Divides it into three equal length intervals. The intermediate S open interval is removed to get two closed interval J1,J2 and I1 = J1 J2. Each of the two resulting output interval are divided into three equal intervals in the same way of I0. The open intermediate interval in each was removed to get four closed S S S intervals J3,J4,J5,J6. Set I2 = J3 J4 J5 J6. Continue this process, to get
n In which is a union of 2 closed intervals obtained by removing the open middle
27 thirds from each of the 2n−1 closed intervals.
3.1 Construction of the Cantor set
Consider the unit interval I0 = [0, 1]. Divide I0 into three equal length intervals
1 2 with end points {0, 3 , 3 , 1}. Remove the open middle third from the interval I0 1 2 S 1 S 2 to get the set I1 = I0 − ( 3 , 3 ) = J1 J2 =[0, 3 ] [ 3 , 1]. Again divide each of the two closed intervals J1 and J2 into three equal length interval with end points
1 2 1 2 7 8 {0, 9 , 9 , 3 , 3 , 9 , 9 , 1}. Remove the middle third from each yield closed interval I2 = 1 2 S 7 8 S S S 1 2 1 2 7 I1 − ( 9 , 9 ) ( 9 , 9 ) = J3 J4 J5 J6. where J3 = [0, 9 ], J4 = [ 9 , 3 ], J5 = [ 3 , 9 ], 8 n J6 = [ 9 , 1] Inductively, we construct a sequence of sets In which is a union of 2 1 T∞ disjoint closed intervals each of them with diameter 3n . Defined the set C = n=1
In.
T∞ Definition 3.1.1. (Kannan, 1979) The set C = n=1 In is called the Cantor set.
If we consider C as a subspace of I with its usual topology, C is called the Cantor space.
Remark 3.1.2. 1) Later we will show that the Cantor set C is nonempty set.
2) The Cantor set is a subset of I formed from deleting accountable union of
open interval.
We will study the properties of this space in the next section. But now, we give some representations equivalent (up to homeomorphic) to the Cantor space. Then,
28 any time and any where, we will use any representation of them as the Cantor space.
3.2 Equivalence representation of the Cantor space
There are many equivalence representations of the Cantor space, and we will give two representations of them.
3.2.1 The sequence representation
Lemma 3.2.1. (Willard, 1970) Any x ∈ [0, 1] has a series expansion of the form ∞ X xi x1 x2 x3 x = 3i = 3 + 32 + 33 + ··· , where xi = 0, 1, 2. i=1
If we express x as (x1, x2, x3, ··· ), we get a sequence representation of x. This representation is unique up to the sequence with ultimately number 2. That is, any sequence which is ultimately 2 can be transformed into other form. For example,
0 2 2 the element x of the sequence representation (0, 2, 2, 2, ··· ) is x = 3 + 32 + 33 +··· = 1 1 1 0 0 3 . But 3 = 3 + 32 + 33 +··· , so it has another sequence representation (1, 0, 0, ··· ). 1 In this case, we consider 3 = (0, 2, 2, ··· ).
Theorem 3.2.2. (Kannan, 1979) An element x ∈ [0, 1] belongs to the Cantor set
C if and only if it has a sequence representation (xi : i ∈ N) where xi = 0, 2. That is, x = (x1, x2, x3, ··· ), where xi 6= 1 ∀ i.
Remark 3.2.3. The Cantor set can be put in 1 − 1 correspondence with the set of
0 0 all sequences of 0 s and 1 s. So by convention, we will replacing each xi = 2 by
29 0 xi = 1 to get an accepted froms of the sequence expansions of the Cantor elements.
Hence we can see the Cantor set C as follows:
C = {(x1, x2, x3, ··· ): xi = 0, 1}.
Example 3.2.4. 1) The element x = (0, 1, 1, 0, 0, 1, 0, 0, 0, ··· ) in C in sequence
2 2 2 expansion is equal to x = 32 + 33 + 36 = 0, 2990397805.
1 1 1 0 0 2 2 2 2) For the element 3 ∈ C, since 3 = 3 + 32 + 33 + ··· = 32 + 33 + 34 + ··· , 1 the original sequence representations of 3 is (1, 0, 0, 0, ··· ) and (0, 2, 2, 2, ··· ). As we mention above, we will consider the original sequence representation
1 1 of 3 as (0, 2, 2, 2, ··· ) and hence the sequence representation of 3 in C equals (0, 1, 1, 1, ··· ).
3) The element x = (1, 0, 0, ··· ) in C in sequence expansion is equal to x =
2 0 0 2 3 + 32 + 33 + ··· = 3 .
Now for each i ∈ N, let Ci = {0, 1}. Then the Cantor set C is exactly the ∞ Y cartesian product of {Ci : i ∈ N}. So C = Ci = {(xi)i∈N : xi ∈ Ci}. Our big i=1 question here is that, the Cantor topology on C as a subspace of I with its standard topology transforms into a topology on C as a countable product of Ci, what is it form? look at the following theorem
Theorem 3.2.5. (Willard, 1970) For any i ∈ N, let Ci = {0, 1} with the discrete topology and let τc be the Cantor topology on C. Then the Cantor topology τc ∞ Y transform into the product topology τp on Ci in the sequence representation of i=1 ∞ Y the Cantor set. Hence, the basic open of the Cantor topology τc has the form Ui i=1 where Ui is open and Ui = {0, 1} for all i except for i = i1, i2, ··· , in.
30 3.2.2 The representation of C as power set of N
Let A ⊆ N and χA : N → {0, 1} be the characteristic function defined as follows: 1, if i ∈ A; χA(i) = 0, if i∈ / A.
Then χA is a sequence in {0,1}. So, χA ∈ {(xi)i∈N : xi = 0, 1} = C. On the other hand, let x = (x1, x2, x3 ··· ) ∈ C. Define A ⊆ N as A = {i : xi = 1}. Then
χA = x ∈ C. That is, any x ∈ C can be considered as a subset A of N where
χA = x. Thus, we can see C as a power set P(N) of the natural numbers with the product topology. in the Cantor set is the induced standard topology on C as a subset of I.
Example 3.2.6. : Let A ={2, 5, 7, 9, 12} in P(N). The sequence representation of
A is x = χA=(0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, ··· ). The value of the Cantor element
2 2 2 2 2 x is x = 32 + 35 + 37 + 39 + 312 = 0, 231472544 ∈ C.
3.3 Properties of the Cantor space
The Cantor space contains some important properties as shown in the following theorems:
Theorem 3.3.1. (Thomson et al., 2008) (Cantor’s intersection theorem) Let X be a Hausdorff space and let (Ck)k∈N be a sequence of nonempty compact subsets of ∞ \ X satisfying C1 ⊇ C2 ⊇ · · · Ck ⊇ Ck+1 ⊇ · · · , then ( Ck) 6= ∅. k=1
31 ∞ \ Proof. Suppose to contrary that Cn = ∅. ∀n ∈ N define Un = C1 − Cn. Then n=1 ∞ ∞ ∞ [ [ \ by DeMorgan’s low, Un = (C1 − Cn) = C1 − (Cn) = C1. Since Un is open n=1 n=1 n=1 in C1 ∀n ∈ N, we get {Un : n ∈ N} is open cover of C1 which is compact. So ∃ k [ finite subcover {un1 , un2 , ··· , unk } such that Uni = C1. Suppose that Unm is the i=1 k [ largest one (choose nm = max{n1, n2, ··· , nk}). Then C1 = Uni = Unm . Thus i=1 Cnm = C1 − Unm = ∅ which is a contradiction to the fact Cn 6= ∅ ∀n.
Lemma 3.3.2. (Willard, 1970) Let R be the set of real numbers with standard topology τst. Then
1. I and R are Hausdorff spaces.
2. A subset A ⊆ R is compact in R iff A is bounded and closed.
Theorem 3.3.3. (Willard, 1970) The Cantor set is a closed subset in [0, 1].
Proof. Through out the construction of the Cantor set, we note that In is a finite ∞ \ union of closed intervals, so it is closed in I = [0, 1]. Now, The Cantor set C = In n=1 is the intersection of closed sets, so it is closed in [0, 1].
Recall that the space [0, 1] is compact. Moreover, a closed subspace of a compact space is compact. This fact give the following Theorem.
Theorem 3.3.4. (Willard, 1970) The Cantor set is a compact Hausdorff space.
Proof. A closed subspace of a compact Hausdorff space is compact and Hausdorff.
32 Remarks 3.3.5. (Willard, 1970)
1. It was proved that a denumerable product of nontrivial finite discrete space
is homeomorphic to the Cantor space, so we consider it as a Cantor space.
2. In the sense of topology, compactness and discreteness are really dual proper-
ties. But the Cantor space is the only infinite space carry the two properties.
Theorem 3.3.6. (Willard, 1970) The Cantor set C is nonempty set.
Proof. I = [0, 1] is a Hausdorff space and In (in the construction of the Cantor space) is nonempty closed and bounded in R, so it is compact. Moreover, I1 ⊇ I2 ⊇ ∞ \ I3 ⊇ · · · . By Cantor’s intersection theorem by Theorem 3.3.1, C = In 6= ∅. n=1 Theorem 3.3.7. (Willard, 1970) The Cantor space is totally disconnected.
Proof. Let A ⊆ C with more than one element, and Let x 6= y in A. Then there exists i ∈ N such that xi 6= yi. So either xi = 0, yi = 1 or xi = 1, yi = 0, say −1 T −1 T xi = 0, yi = 1. Take U = πi ({0}) A and V = πi ({1}) A. Then both U and V are open in A, U T V = ∅ and x ∈ U, y ∈ V . That is, A is disconnect subset of
C.
Recall that a set A in a space X is perfect in X if A is closed and dense in itself
(:= each point of A is an accumulation point of A). This compact is give in the following example:
Example 3.3.8. (Willard, 1970) Let R be the set of real numbers with standard topology.
33 1. If A=[0, 1], then A is closed and A0 =A. So A is perfect.
2. If B=[0, 1] S{2}, then B is closed, and B0 = [0, 1] 6= B. Hence B is not
perfect.
0 0 3. If C = [0, 1), then C is not closed and C = [0, 1]. Although C ⊆ C , C is not perfect.
0 4. If Q is the set of rational number, then Q ⊆ Q = R. Since Q is not closed
so Q is not perfect in R.
S 1 5. (Q, τst) is perfect space while (A, τst) is not perfect, where A = {0} { n : n ∈ N}.
6. The space (R, τst) is perfect space.
7. Consider the cofinite space (R, τcof ), the only perfect set in R is R it self.
Theorem 3.3.9. (Willard, 1970) The Cantor space C is a perfect space.
−1 Proof. Let x = (x1, x2, ··· ) ∈ C and suppose that U = πi (ui) is a subbasic neighborhood of x where i ∈ N and ui ∈ τi =τdisc. Then xi ∈ ui. Suppose i 6= 1,
0 0 0 and define x1=1 if x1 = 0 and x1 = 0 if x1 = 1. If set y=(x1, x2, x3, ··· ), then we
−1 0 0 get y 6= x in C and y ∈ πi (ui). This proves that x ∈ C and C ⊆ C .
Definition 3.3.10. (Willard, 1970) Let M be a nonempty set and ρ : M × M −→
R. Then ρ is called a metric function on M if it satisfies the following:
1. d(x, y) ≥ 0 (Non-negativity).
2. d(x, y) = 0 ⇔ x = y (identity of indiscernibles).
34 3. d(x, y) = d(y, x) (symmetry).
4. d(x, z) ≤ d(x, y) + d(y, z) ( triangle inequality).
The pair (M, ρ) is called a metric space.
Definition 3.3.11. (Willard, 1970) Let x ∈ M and > 0. Define Uρ(x, ) = {y :
ρ(x, y) < }. Then Uρ(x, ) is called a ball of x.
Definition 3.3.12. (Willard, 1970) A subset U ⊆ M is called open if ∀x ∈ U,
∃ > 0, such that Uρ(x, ) ⊆ U.
Theorem 3.3.13. (Willard, 1970) If (M, ρ) is a metric space and if τρ = {U ⊆
M : Uis open}, then τρ forms a topology on M called a metric topology on M induced by ρ.
Definition 3.3.14. (Willard, 1970) If (X, τ) is a topological space, if there exists a metric ρ on X such that τρ = τ, then (X, τ) is called a metrizable topology.
Remark 3.3.15. (Willard, 1970) It is well known that not all topologies are metriz- able. When X is metrizable, and if ρ is a metric on X such that τρ = τ then τ2ρ = τ
+ and more general, τkρ = τ for any constant k ∈ R where (kρ)(x, y) = kρ(x, y). Therefor when (X, τ) is metrizable, then there exists infinity many metric on X generate τ.
Theorem 3.3.16. (Willard, 1970) A subset of a metrizable space is metrizable.
Proof. If ρ : X × X −→ R is a metric generates τ on X and if A ⊆ X, then the restrected function ρ|A×A : A × A −→ R is a metric on A generates τA.
35 Example 3.3.17. (Willard, 1970) The standard topology τst on R is metrizable.
The metric ρ on R defined as ρ(x, y) = |x − y| generates the standard topology. So, the space (I, τst) is metrizable since I is a subspace of R.
Example 3.3.18. (Willard, 1970) Any discrete space is metrizable, with a metric
ρ defined as
0, x = y, ρ(x, y) = 1, x 6= y. Theorem 3.3.19. (Willard, 1970) A product space is metrizable iff each factor is metrizable and there are ≤ ℵ0 factors.
Theorem 3.3.20. (Willard, 1970) The Cantor space 2N is metrizable spaces.
Proof. Since {0, 1} with discrete topology is metrizable, and since N is countable, 2N is countable product of metrizable space. By Theorem 3.3.19, 2N is metrizable.
Theorem 3.3.21. (Willard, 1970) All totally disconnected, perfect compact metric spaces are homeomorphic.
Corollary 3.3.22. (Willard, 1970) The Cantor set is the only totally disconnected, perfect compact metric space (up to homeomorphism).
Proof. By Theorem 3.3.7, C is totaly disconnected. Moreover, by Theorem 3.3.4, C is compact Hausdorff space. Also, by Theorem 3.3.9, C is perfect. Since I with the standard topology is metrizable and C is a subspace of I, then by Theorem 3.3.20,
C is metrizable. So by Theorem 3.3.21, the Cantor set is only totally disconnected (up to homeomorphism).
36 Theorem 3.3.23. (Willard, 1970) A product space is totally disconnected iff each factor space is totally disconnected.
Corollary 3.3.24. (Willard, 1970) The Cantor set C is homeomorphic to CN.
37 Chapter 4
The Cantor Cube of Infinite set X Chapter 4
The Cantor Cube of Infinite set X
4.1 The Cantor cube 2X
For the Cantor space, we attempt to a systematic study of analytic topologies over the natural numbers N (or any countable set X). We identify every subset of N with its characteristic function. This implies that the power set P(N) is identified with the Cantor space 2N. Now and for more general, let X be an infinite set P(X) the power set of X and 2X = {0, 1}X the collection of all functions from X to
{0, 1}. If ∀α ∈ X, we define Xα = {0, 1} with discrete topology τα, then the set
X Y 2 = Xα will be the product space of the collection {Xα : α ∈ X}. In this α∈X case, the space 2X is called a Cantor cube of X. Now, let A ⊆ X and defined 1, if x ∈ A; χA(x) = 0, if x∈ / A.
39 X X Then χA ∈ 2 . On the other hand, if f ∈ 2 , define B = {x ∈ X : f(x) = 1}.
X X Then χB = f. Hence there is a bijection φ from P(X) and 2 φ : P(X) −→ 2 defined as φ(A) = χA. We can use this bijection to introduce a topology on
X X P(X) as follows: If τp is the product topology on 2 and if T is open in 2 then φ−1(T ) is open in P(X). In this case P(X) is homeomorphic to 2X and φ is a homeomorphism between them. For this end, as we do in the Cantor space, we can consider P(X) to be the Cantor cube with the product topology on P(X). From now on, and for any infinite set X (with topology or with out one), the power set
P(X) as a space will be consider as the Cantor cube 2X with the product topology
X τp. Moreover, P(X) can be identify as 2 without any mention. The element
X X A ∈ P(X) will identify as φ(A) = χA in 2 and the element f in 2 will identify as φ−1(f) = B = {x ∈ X : f(x) = 1} in P(X).
Definition 4.1.1. (Uzc´ategui, 2005) Let X be an infinite set and let P(X) be power set of X If K,F ∈ P(X), we defined the interval [K,F ] = {A ∈ P(X):
K ⊆ A ⊆ F }.
Theorem 4.1.2. (Uzc´ategui, 2005) Let X be an infinite set and P(X) the Cantor cube of X. Then the collection B = {[K,F ]: K is finite and F cofinito} is a base for the product topology on P(X).
X Proof. Clearly, if U is a basic open in τp on 2 then ∃ α1, α2, ··· , αn in X such that
−1 \ \ −1 Y U = πα1 (Uα1 ) ··· παn (Uαn ) = Uα α∈X −1 Where Uα = Xα = {0, 1} ∀α 6= α1, ··· , αn and Uαi ∈ ταi . The open set φ (U) will
be basic open for τp on P(X). Suppose without loss of generality that Uα1 = Uα2 =
40 ··· = Uαr = {1} and Uαr+1 = Uαr+2 = ··· = Uαn = {0}. Set K = {α1, α2, ··· , αr} and F = X − {αr+1, αr+2, ··· , αn}. Then K is finite and F is cofinite subset of X.
−1 Claim: φ (U) = [K,F ]. [To see this, let f ∈ U. Then f(αi) ∈ Uαi for i =
1, 2, ··· , n. So, f(αi) = 1 ∀i = 1, 2, ··· , r and f(αi) = 0 ∀i = r + 1, r + 2, ··· , n.
−1 If B = φ (f) = {x ∈ X : f(x) = 1}, then αi ∈ B ∀i = 1, 2, ··· , r. Hence K ⊆ B.
Moreover, αi ∈/ B ∀i = r + 1, r + 2, ··· , n so B ⊆ F . Therefore B ∈ [K,F ], and
−1 φ (U) ⊆ [F,K]. On the other hand, if K ⊆ A ⊆ F , then αi ∈ A ∀i = 1, 2, ··· , r and αi ∈/ A ∀i = r + 1, r + 2, ··· , n. Hence 1, if i = 1, 2, ··· , r φ(A)(αi) = χA(αi) = 0, if i = r + 1, r + 2, ··· , n.
−1 So, ∀i = 1, 2, ··· , n, φ(A)(αi) ∈ Uαi and hence φ(A) ∈ U. Thus, A ∈ φ (U) and [K,F ] ⊆ φ−1(U)]. By proving this claim, we get the result.
Theorem 4.1.3. (Uzc´ategui, 2005) Let X be an infinite set and 2X = P(X) the
Cantor cube of X. Then for any subsets K and F of X where K finite F cofinite the basic open set [K,F ] in the product topology is both open and closed.
Proof. Firstly, [K,F ] is basic open and hence open set. Now,
[ [ [ P(X) − [F,K] = [{l},X] [∅,X − {k}] l∈X−F k∈K is a union of basic open sets of the form [{l},X], l ∈ X − F and [∅,X − {K}], k ∈ K. So, it is Open and hence [K,F ] is closed set in P(X).
41 4.2 Properties of different types of cube
Definition 4.2.1. (Engelking, 1997) Let I=[0,1] with the standard topology. If
A is nonempty indexed set, then the product space IA is called a cube . If A is
A ℵ0 denumerable (when|A| = ℵ0), then I (also written I ) is called the Hilbert cube.
Recall that a space X is second countable if it has a countable base. X is separable if it has a countable dense. And X is Lindel¨of if any open cover of X has a countable subcover.
Theorem 4.2.2. (Willard, 1970) A compact Hausdorff space is a T4-space.
Theorem 4.2.3. (Willard, 1970) If X is second countable, then X is Lindelo¨f and separable.
Theorem 4.2.4. (Willard, 1970)
1. A product of Hausdorff space, each with at least two point, is separable iff
each factor is separable and there are ≤ c factors.
2. A product of Hausdorff space is second countable iff each factor is second
countable and there are ≤ ℵ0 factors.
Theorem 4.2.5. Let X be an infinite set, A a nonempty set, and N the set of natural numbers. Then the Cantor space 2N, the Cantor cube 2X and the cube IA are compact Hausdorff spaces.
Proof. Since {0, 1} with discrete topology is compact Hausdorff space, so by The- orem 1.3.7 and Theorem 1.3.6, they are compact Hausdorff spaces. Similarly, since
42 I = [0, 1] is compact Hausdorff, then the cube IA is a product of compact Hausdorff space so it is compact.
Corollary 4.2.6. The Cantor space 2N, the Cantor cube 2X and the cube IA are all T4-spaces.
Proof. Direct from Theorem 4.2.2 and Theorem 4.2.5.
Theorem 4.2.7. Let X be an infinite set and A a nonempty set. Then
X 1. The Cantor cube 2 is second countable iff X is countable (|X| ≤ ℵ0).
2. The cube IA is second countable iff A is countable.
Proof. Since {0, 1} and I are second countable and Hausdorff spaces, and 2X and
IA are product of Hausdorff spaces each factor is second countable and there are
|X| of factors of 2X and |A| of factors of IA, the result then come directly from
Theorem 4.2.4 point(2).
Corollary 4.2.8. The Cantor space 2N and the Hilbert cube IN are second count- able, and then separable and Lindelo¨f.
Theorem 4.2.9. Let X be an infinite set and A a nonempty set. Then
1. The Cantor cube 2X is separable iff |X| ≤ c.
2. The cube IA is separable iff |A| ≤ c.
Proof. Similar to the proof of previous theorem.
Recall that A product space is metrizable iff each factor is metrizable and there are ≤ ℵ0 factors. This fact gives the following Theorem:
43 Corollary 4.2.10. The Cantor space 2N and The Hilbert cube IA are metrizable spaces.
Proof. Since {0, 1} with discrete topology and I = [0, 1] with standard topology are metrizable, and since N is countable, 2N and IN are countable product of metrizable space. By Theorem 3.3.19, they are metrizable.
Corollary 4.2.11. Let X be an infinite set and A a non-empty set. Then the
X A Cantor cube 2 (the cube I ) is metrizable iff |X| ≤ ℵ0 (|A| ≤ ℵ0).
Proof. Replace N in the proof of previous corollary by X and A.
Theorem 4.2.12. Let (X, τst) be a metric space. Then X is homeomorphic to a subspace of the cube IX .
4.3 More on the Cantor cube 2X
Theorem 4.3.1. (Todorcevic and Uzc´ategui, 2001) Let τ be a topology on X. The following statements are equivalent:
1. τ is Alexandroff topology on X.
2. τ is closed subset in 2X .
X Proof. (1 ⇒ 2) Let {Aα}α∈∆ be a net of open in τ that converges to A (in 2 ). By Lemma 1.2.4 it suffices to prove that A ∈ τ. So x ∈ A. Since τ is Alexandroff, there exists a minimal nhood Nx of x.
Claim: Nx ⊆ A, and hence A is open in X. [To see this, let p∈ / A. Then
44 {x} ⊆ A ⊆ X − {p}. Hence [{x},X − {p}] is a basic nhood of A in 2X . Since
Aα −→ A, ∃α0 ∈ ∆ such that ∀α ≥ α0, we get Aα ∈ [{x},X − {p}]. So, ∀α ≥ α0, x ∈ Aα and Aα ⊆ X − {p}. Now Aα ∈ τ, so Nx ⊆ Aα and p∈ / Aα ∀α ≥ α0. Thus
X p∈ / Nx. This proves that Nx ⊆ A and so A is open in X and so τ is closed in 2 ].
(2 ⇒ 1) Let x ∈ X and Ux be the nhood system of x in X. Take. ∆ = {U ∈ Ux : U ∈ τ} and defined a relation ≤ on ∆ as follows: U ≤ V in ∆ iff V ⊆ U. Then ∆
X is directed set. Let ν : ∆ −→ 2 be defined as ν(u) = U. Then (νu)u∈∆ is a net in
X \ τ in 2 (The element νu are all open in X). Set Nx = U. U∈∆ X X Claim: ν → Nx in 2 . [To see this, let [K,F ] be any basic nhood of Nx in 2 .
Then K and X − F are finite subsets of X and K ⊆ Nx ⊆ F . So, X − F ⊆
X − Nx ⊆ X − K. Suppose that X − F = {x1, x2, ··· , xn}. Then ∀i = 1, 2, ··· , n, [ xi ∈ X − F ⊆ X − Nx = (X − U). Thus, ∃Ui ∈ ∆ such that xi ∈ X − Ui for U∈∆ n n [ Tn \ i = 1, 2, ··· , n. Hence X − F ⊆ X − Ui and so i=1 Ui ⊆ F . Choose U0 = Ui i=1 i=1 \ in ∆. If V ≥ U0, then V ⊆ U0 ⊆ F . Now if r ∈ K ⊆ NX then r ∈ U ⊆ V (since U∈∆ V ∈ ∆). This proves that K ⊆ V . Therefor K ⊆ V ⊆ F . Thus νv = V ∈ [K,F ].
∀V ≥ V0].
X X Now (νu)u∈∆ is a net in τ in 2 that converges to Nx and τ is closed in 2 , by
Lemma 1.2.4, Nx ∈ τ. Thus, x has a minimal open nhood and by Theorem 2.2.2, X is Alexandroff space.
Lemma 4.3.2. (Uzc´ategui, 2005) Let X be an infinite set and 2X the Cantor cube of X. Let f, g : 2X × 2X −→ 2X defined as: f(A, B) = A T B, and g(A, B) =
A S B. Then f and g are continuous function.
45 Proof. To prove that f is continuous, let (A, B) ∈ 2X × 2X and V = [K,F ] be a neighborhood of A T B. Then K ⊆ A T B ⊆ F . Hence K ⊆ A, K ⊆ B.
Take W = F S A and N = B S F . So K ⊆ A ⊆ W , K ⊆ B ⊆ N. If we take
U = [K,W ] × [K,N], then U is open in 2X × 2X . We want prove f(U) ⊆ V , so let G ∈ f(U). Then there exists (S,D) ∈ 2X × 2X such that G = S T D. Hence
K ⊆ S ⊆ W and K ⊆ D ⊆ N. This implies that K ⊆ S T D = G ⊆ W T N =
F S(A T B) ⊆ F S F = F . Therefore G ∈ [K,F ] and hence f(U) ⊆ V . This proves that f is continuous.
To prove that g is continuous, let (A, B) ∈ 2X × 2X and V = [K,F ] be a neigh- borhood of A S B. Then K ⊆ A S B ⊆ F . So A ⊆ F and B ⊆ F . Take
W = K T A and N = B T K. Then W ⊆ A ⊆ F and N ⊆ B ⊆ F . If we take
U = [K,W ] × [K,N], then U is open in 2X × 2X . We want prove g(U) ⊆ V , so let G ∈ f(U). Then there exists (S,D) ∈ 2X × 2X such that G = S S D. Hence
W ⊆ S ⊆ F and N ⊆ D ⊆ F . This implies that W S N ⊆ S S D = G ⊆ F S F =
F , and hence so K = K T K ⊆ K T A S B = W S N ⊆ G ⊆ F . Thus G ∈ [K,F ] and g(U) ⊆ V . Therefore g is continuous.
Theorem 4.3.3. (Uzc´ategui, 2005) Let X be an infinite set and 2X the Cantor cube of X. If f : 2X −→ 2X is defined by f(A) = X − A, then f is a homeomorphism.
Proof. Clearly f is 1 − 1 and onto function. Let [K,F ] be a basic open in 2X , then f[K,F ] = {f(A): K ⊆ A ⊆ F } = {X − A : X − F ⊆ X − A ⊆ X − K} =
[X − F,X − K] (If K ⊆ A ⊆ F then X − F ⊆ X − A ⊆ X − K). Since X − F is finite and X − K is cofinite. We get f[K,F ] = [X − F,X − K] is open in 2X .
Similarly, f −1[K,F ] = [X − F,X − K] is open. Then f is open and continuous and
46 hence a homeomorphism.
Theorem 4.3.4. (Uzc´ategui, 2005) Let F be closed subset of 2X . Then
1. If F is closed under finite intersection, then Fis closed under arbitrary in-
tersection.
2. If F is closed under finite union, then F is closed under arbitrary union.
Proof. 1. Let {Aα}α∈∆ be a collection of subsets in F, and let K = {ν ⊆ ∆ : ν
is finite}. Define a relation ≤ on K as follows: A1 ≤ A2 if A1 ⊆ A2 (the usual inclusion). Then (K, ≤) is directed set. Let C : K → 2X be defined as
follows: \ C(ν) = Aα α∈ν X Then (Cν)ν∈∆ is a net in 2 , and ∀ν ∈ ∆, Cν is a finite intersection of element
X in F, so it is in F. Thus (Cν)ν∈∆ is a net in F which is a closed in 2 . By
Lemma 1.2.4 if Cν → C, then C must be in F. \ Claim: Cν → Aα [To see this, suppose that [K,F ] be a basic open of α∈∆ \ X \ [ { Aα in 2 . Then K ⊆ Aα ⊆ F . So X − F ⊆ Aα. Suppose that α∈∆ α∈∆ α∈∆ X − F = {x , x , ··· , x }, then ∃α , α , ··· , α such that x ∈ A{ and hence 1 2 n 1 2 n i αi n n [ { \ X −F ⊆ Aα. By DeMorgan’s low, Aαi ⊆ F . Take ν0 = {α1, α2, ··· , αn} i=1 i=1 n \ \ \ in K. If ν ≥ ν0, then ν ⊇ ν0. So Cν ⊆ Aα ⊆ Aα = Aαi ⊆ F . Thus α∈∆ α∈ν0 i=1 \ \ we have K ⊆ Aα ⊆ Aα = Cν ⊆ F . So Cν ∈ [K,F ] ∀ν ≥ ν0. Hence α∈∆ α∈ν \ Cν → Aα ∈ F and F is closed under arbitrary intersection]. α∈∆
47 2. The proof of this part is nearly similar to the proof of part (1) as follows:
Let {Bα}α∈∆ be a collection of subsets in F and K = {ν ⊆ ∆ : ν is finit}.
Suppose that K is directed as in part (1): A1 ≤ A2 in K if A1 ⊆ A2. Then
X [ X D : K → 2 defined as D(ν) = Aα is a net in F in 2 . Since F is closed α∈ν X in 2 , if Dν → D then D ∈ F. [ Claim: Dν → Aα [To see this, suppose that [K,F ] is a basic nhood α∈∆ [ X [ of Aα in 2 , then K ⊆ Aα ⊆ F . Since K is finite, say K = α∈∆ α∈∆ n [ {y1, y2, ··· , yn}. Then ∀i{1, 2, ··· , n}, ∃αi ∈ ∆ such that K ⊆ Aαi . i=1 n [ Take ν0 = {α1, α2, ··· , αn}. If ν ≥ ν0, then ν ⊇ ν0 and so K ⊆ Aα ⊆ i=1 [ [ Aα = Dν ⊆ Aα ⊆ F . Hence Dν ∈ [K,F ] ∀ν ≥ ν0. This implies that α∈ν α∈∆ [ Dν → Aα ∈ F and F is closed under arbitrary union]. α∈D
Remark 4.3.5. If (∆1, ≤1) and (∆2, ≤2) are two directed sets and if ∆1 × ∆2 is the cartesian product, then we can define a direction ≤c an ∆1 × ∆2 as follows:
(α1, β1) ≤c (α2, β2) iff α1 ≤1 β1 and α2 ≤2 β2. This direction is called coordinate wise direction .
Theorem 4.3.6. (Uzc´ategui, 2005) Let τ be a topology on X. Then the closure τ of τ in 2X is Alexandroff topology on X.
Proof. By Theorem 4.3.4, it is enough to show that τ is closed under finite inter- section and finite union. Let A, B ∈ τ. Then by Theorem 1.2.5 there are two net
{Aα}α∈∆1 , {Bβ}β∈∆2 in τ such that Aα → A and Bβ → B. Let ∆ = ∆1 × ∆2
48 x X be the coordinate wise direction of ∆1 × ∆2. Define P : ∆ → 2 × 2 by
X X P (α, β) = (Aα,Bβ). Then (Aα,Bβ)(α,β)∈∆ is a net in 2 × 2 .
Claim:(Aα,Bβ) → (A, B) [To see this, suppose that U × V is a basic open of
X X (A, B) in 2 × 2 . Then U is a nhood of A and V is nhood of B. so, ∃α0 ∈ ∆1,
β0 ∈ ∆2 such that ∀α ≥ α0, Aα ∈ U and ∀β ≥ β0, Bβ ∈ V . Thus ∀(α, β) ≥ (α0, β0)
X X in ∆, we get α ≥ α0 and β ≥ β0 and so (Aα,Bβ) ∈ U × V in 2 × 2 ]. Define f, g : 2X × 2X → 2X as follows: f(C,D) = C T D and g(C,D) = C S D. So, by
X X Lemma 4.3.2, f, g are continuous Since (Aα,Bβ) → (A, B) in 2 ×2 , by continuity of f and g and by Theorem 1.2.6 f(Aα,Bβ) → f(A, B) and g(Aα,Bβ) → g(A, B), T T S Equivalently, the two net (Aα Bβ)(α,β)∈∆ → (A B) and (Aα Bβ)(α,β)∈∆ → S T S (A B). Since Aα,Bβ are open in X, then Aα Bβ ∈ τ and Aα Bβ ∈ τ. By Theorem 1.2.5, A T B ∈ τ and A S B ∈ τ in 2X . Thus τ is closed under finite intersection and finite union By Theorem 4.3.4, τ is closed under arbitrary intersec- tion and arbitrary union and containing ∅, X (since τ ⊆ τ). Thus, τ is Alexandroff topology on X.
Theorem 4.3.7. If (X, τ) is topological space, then the closure τ of τ in 2X is the smallest Alexandroff topology on X containing τ.
Proof. Suppose τ1 is an Alexandroff topology on X containing τ, then τ1 is closed
X X subset in 2 . Since τ is smallest closed subset in 2 containing τ, then τ ⊆ τ1
Definition 4.3.8. Let X be a topological space. Defined ≤τ on X as a ≤τ b if a ∈ {b} which is called pre order on X induced by τ
Lemma 4.3.9. Let X be topological space, then the relation ≤τ is reflexive and transitive but not antisymmetric
49 Proof. it is clear reflexive x ∈ {x}. let x ≤τ y so x ∈ {y} and y ≤τ z so y ∈ {z} then {x} ⊆ y and {y} ⊆ {z} since {x} is the smallest closed that contains x and
{y} is the smallest closed that contain y then {x} ⊆ {z} and so x ∈ {z} this mean that transitive.
Theorem 4.3.10. Let τ and ρ are topological space over X. If τ ⊆ ρ then ≤τ ⊇≤ρ
Proof. suppose that x τ y there exist O ∈ τ such that O contain x and y∈ / O since τ ⊆ ρ. So O ∈ ρ therefore x ρ y. Hence ≤τ ⊇≤ρ
Theorem 4.3.11. (Uzc´ategui, 2005) Let τ be a topology over X, then ≤τ =≤τ , where τ is the closure of τ in cube 2X
Proof. As τ ⊆ τ, by Theorem 4.3.10 ≤τ ⊆≤τ . If x τ y and there exists U such that x ∈ U and y∈ / U.[{x}, {y}{] is a neighborhood of U. As U ∈ τ,[{x}, {y}{] T τ 6= ∅,
{ T then there is G ∈ [{x}, {y} ] τ such that x ∈ G and y∈ / G therefore x τ y, hence ≤τ ⊆≤τ
Corollary 4.3.12. (Todorcevic and Uzc´ategui, 2001) Let τ be topology over X.
Then
1. τ is T0 if and only if τ is T0
X 2. τ is T1 if and only if τ is dens in 2
Proof. (⇒) Let τ be T0 and let x, y ∈ X such that x 6= y. Then there exist Ux ∈ τ such that y∈ / Ux or Vy ∈ τ such that x∈ / Vy. Now Ux ∈ τ or Vy ∈ τ, since τ ⊆ τ, hence τ is T0.
(⇒) Suppose that τ is a T0 and x, y ∈ X such that x 6= y. Then x τ y by
50 Theorem 4.3.11 x τ y. So there exists Nx ∈ τ containing x such that y∈ / Nx or y τ x, Ny ∈ τ containing y such that x∈ / Ny hence τ is T0.
(⇒) Assume that τ is a T1. Let x and y ∈ X such that x 6= y. Then there exists
Ux and Vy ∈ τ ⊆ τ containing x and y respectively such that x∈ / Vy and y ∈ Ux.
X Therefore τ is T1. By Theorem 4.3.1 τ is Alexandroff, so τ = 2 .
X (⇐) Assume that τ is dense so τ = 2 . Then τ is T1. Let x and y ∈ X such that x 6= y so x τ y and y τ x. By Theorem4.3.10 x τ y and y τ x. Then there exist Nx and Nx ∈ τ containing x and y respectively such that y∈ / Nx and x∈ / Ny, hence τ is T1
Theorem 4.3.13. (Todorcevic and Uzc´ategui, 2001) Let τ be a topology over X.
Then the following statements are equivalent:
1. τ is open in 2X .
2. τ is clopen in 2X .
3. ∅,X are interior point of τ in 2X .
4. There is a finite S set such that S is clopen and X − S is τ-discrete.
Proof. (1 ⇒ 3) Assume that τ is open. So τ is a neighborhood of ∅ and X. Hence
∅ and X are interior point of τ.
(3 ⇒ 4) Let ∅ and X be interior points of τ in 2X . There are finite set K and cofinite set F such that [∅,F ] ⊆ τ and [K,X] ⊆ τ. Assume that S = M S K where M = F { then S,S{ ∈ τ such that M S K ∈ [K,X] and S{ ∈ [∅,F ]. Then
{ { S is clopen. Let τS{ be the subspace topology of S and y ∈ S , then y ∈ F and
51 y ∈ K{, so {y} ⊆ F , therefore {y} ∈ [∅,F ] ⊆ τ. So {y} ∈ τ which means that it is open in τS{. Hence X − S is discrete. (4 ⇒ 2) Let S,S{ ∈ τ with S finite such that S{ is discrete subspace of X then
{ τS{ ⊆ τ where τS{ is the discrete topology in S for any k ⊆ S with K ∈ τ, be the basic clopen
X \ UK = [K,F ] = {A ∈ 2 : K ⊆ A, A (S − K)} where F = (S − K){ show that.
[ τ = {UK : K ⊆ S,K ∈ τ} (4.3.1)
T T { Let A ∈ UK then A (S − K) = S (A − K) = ∅. Then A − K ∈ S and therefore S T A − K ∈ τ. So (A − K) K = A ∈ τ. Hence UK ⊆ τ. Let A ∈ τ and K = A S, T K ⊆ A and A (S − K) = ∅ then A ∈ UK . Hence τ ⊆ UK which mean that equation 4.3.1 holds. Since S is finite, τ is a finite union of clopen UK . Therefore τ is clopen in 2X .
(2 ⇒ 1) it is clear is hold
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