Lecture - 10: The Order Topology - · PDF fileIntroduction Order Sets Order Topology Examples...

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Introduction Order Sets Order Topology Examples of order Topology Lecture - 10: The Order Topology Dr. Sanjay Mishra Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Sanjay Mishra The Order Topology

Transcript of Lecture - 10: The Order Topology - · PDF fileIntroduction Order Sets Order Topology Examples...

Page 1: Lecture - 10: The Order Topology - · PDF fileIntroduction Order Sets Order Topology Examples of order Topology Lecture - 10: The Order Topology Dr. Sanjay Mishra Department of Mathematics

IntroductionOrder Sets

Order TopologyExamples of order Topology

Lecture - 10: The Order Topology

Dr. Sanjay Mishra

Department of MathematicsLovely Professional University

Punjab, India

October 18, 2014

Sanjay Mishra The Order Topology

Page 2: Lecture - 10: The Order Topology - · PDF fileIntroduction Order Sets Order Topology Examples of order Topology Lecture - 10: The Order Topology Dr. Sanjay Mishra Department of Mathematics

IntroductionOrder Sets

Order TopologyExamples of order Topology

Outline

1 Introduction

2 Order Sets

3 Order Topology

4 Examples of order Topology

Sanjay Mishra The Order Topology

Page 3: Lecture - 10: The Order Topology - · PDF fileIntroduction Order Sets Order Topology Examples of order Topology Lecture - 10: The Order Topology Dr. Sanjay Mishra Department of Mathematics

IntroductionOrder Sets

Order TopologyExamples of order Topology

Introduction I

Order theory is a branch of mathematics which investigates our intuitivenotion of order using binary relations. It provides a formal framework fordescribing statements such as “this is less than that” or “this precedesthat”. This section introduces basic definitions and their consequence.

Orders are everywhere in mathematics and related fields like computerscience. The first order often discussed in primary school is the stan-dard order on the natural numbers e.g. “2 is less than 3”, “10 is greaterthan 5”, or “Does Tom have fewer cookies than Sally?”. This intuitiveconcept can be extended to orders on other sets of numbers, such as theintegers and the reals. The idea of being greater than or less than an-other number is one of the basic intuitions of number systems. Another

Sanjay Mishra The Order Topology

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Order TopologyExamples of order Topology

Introduction II

familiar example of an ordering is the ”lexicographic order” of words ina dictionary.Types of orders have a special property: each element can be comparedto any other element, i.e. it is greater, smaller, or equal. However, thisis not always a desired requirement. For example, consider the subsetordering of sets. If a set A contains all the elements of a set B, then Bis said to be smaller than or equal to A. Yet there are some sets thatcannot be related in this fashion. Whenever both contain some elementsthat are not in the other, the two sets are not related by subset-inclusion.Hence, subset-inclusion is only a partial order, as opposed to the totalorders given before.

Sanjay Mishra The Order Topology

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Order Sets I

Definition (Partial ordered sets)

A binary relation “≤” is said to partial order if for all x, y and z of A ifhold these three conditions :

1 (Reflexive): For all x in A, x ≤ x.

2 (Antisymmetric): For all x, y in A, x ≤ y, y ≤ x⇒ y = x.

3 (Transitive): For all x, y, z in A, x ≤ y, y ≤ z ⇒ x ≤ z

In this case the set A is called a partially ordered set and is denoted bythe symbols (A,≤).

Sanjay Mishra The Order Topology

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Order Sets II

Example

The relation on N defined by x < y is a partial order. This order iscalled natural order or usual order.

Sanjay Mishra The Order Topology

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Order Sets III

Example

Let A be a family of sets. The relation in A defined by the opensentence “A is a subset of B” is a partial order. This relation in A iscalled “set inclusion” relation.

Sanjay Mishra The Order Topology

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Order Sets IV

Remark

If a relation R is a partial order in A, then the inverse relation R−1 isalso a partial order in R.

Sanjay Mishra The Order Topology

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Order Sets V

Definition

Any two elements x, y in a partial order set (A,≤) are said to becomparable if either x ≤ y or y ≤ x. And said to uncomparable ifx ≤ y and y ≤ x both are not true.

Example

Let A = {1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 18}. A relation < defined in A byx < y if and only if 3x = y. Here the set (A,<) is partial order set.Two elements 1 and 2 are uncomparable because a 3 · 1 6= 2. but 2 and6 are comparable since 3 · 2 = 6.

Sanjay Mishra The Order Topology

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Order Sets VI

In partial ordered set some elements are possible to uncomparable. Nowthink about such set in which all elements are comparable.

Definition (Totally ordered sets)

If every pair of elements of partial order set is comparable, then suchset is called totally ordered set, or linearly ordered set, or chain.

Example

Let A = {2, 6, 18, 24, 72}. A relation < on a A is defined as x < y ifand only if x divides y. Here every pari of elements of partial order setis comparable, hence (A,<) is totally ordered set.

Sanjay Mishra The Order Topology

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Order Sets VII

Definition (Simple order Set)

A relation C on a set A is called an simple order set if it has hold threeproperties as like Comparability, Non-reflexivity and Transitivity.

Sanjay Mishra The Order Topology

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Order Sets VIII

Definition (Subset of an ordered set)

Let (A,<) be partial order set and B ⊂ A be arbitrary, then partialorder R in A induces a partial order R′ in B such that

aRb; a, b ∈ B ⇒ aR′b

And we say that (B,R′) is a subset of a partial order set (A,R).

Remark

Similarly we can define the subset of totally ordered set. A subset ofpartial order set and totally order set is partial and totally ordered setrespectively.

Sanjay Mishra The Order Topology

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Order Sets IX

Definition (Bounds of an ordered set)

Let (X,≤) be a partial order set and (A,≤) be the subset be its subset.An element a of X is said to be an upper bound of A if x ≤ a for all xof A.An upper bound b of A is called a least upper bound or supremum of Aif b ≤ a for every upper bound a of A. And its denoted by sup(A).An element a of X is said to a lower bound of A if for all x ∈ A, a ≤ x.

Sanjay Mishra The Order Topology

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Order Topology I

If X is a simply ordered set, there is a standard topology for X, definedusing the order relation. It is called the order topology. Here we willconsider it and study some of its properties.Suppose that X is set having a simple order relation <. Given elementsa and b of X such that a < b, there are four subsets of X that are calledthe intervals determined by a and b. They are following ;

(a, b) = {x : a < x < b}, (a, b] = {x : a < x ≤ b},

[a, b) = {x : a ≤ x < b}, [a, b] = {x : a ≤ x ≤ b}

Sanjay Mishra The Order Topology

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Order Topology II

Definition

Let X be a set with a simple order relation and assume that it hasmore than one element. Let B be the collection of all subsets of thefollowing types :

1 All open intervals (a, b) in X.

2 All intervals of the form [a0, b),where a0 is the smallest element (ifany) of X.

3 All intervals of the form (a, b0],where b0 is the largest element (ifany) of X.

The collection B is a basis for a topology on X, which is called theorder topology.

Sanjay Mishra The Order Topology

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Order Topology III

Remark

If X has not smallest element, there are not sets of type Go (2), andif X has not largest element, there are not sets of type Go (3).

Sanjay Mishra The Order Topology

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Examples of order Topology I

Example

The standard topology on R is the order topology derived from theusual/standard order on R.

Sanjay Mishra The Order Topology

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Examples of order Topology II

Example

The positive integer Z+ form an ordered set with a smallest elementand it is discrete topology.

Sanjay Mishra The Order Topology

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Examples of order Topology III

Let X = {5} ∪ (0, 1) ∪ [4, 8). A basis for the order topology on X:

1 All open intervals (a, b) in X.

(a, b) =

(0, b), if a = 5, b2(0, 1);

(0, 1), if a = 5, b = 4;

(0, 1) ∪ [4, b), if a = 5, b ∈ (4, 8);

(a, b), if a ∈ (0, 1), b ∈ (0, 1);

(a, 1), if a ∈ (0, 1), b = 4;

(a, 1) ∪ [4, b), if a ∈ (0, 1), b ∈ (4, 8);

(a, b), if a ∈ [4, 8), b ∈ (4, 8).

Sanjay Mishra The Order Topology

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Examples of order Topology IV

2 All intervals of the form [a0, b), where a0 is the smallest element (ifany) of X.

[−5, b) =

{5} ∪ (0, b), if b ∈ (0, 1);

{5} ∪ (0, 1), if a = 5, b = 4;

{5} ∪ (0, 1) ∪ [4, b), if a = 5, b ∈ (4, 8).

3 All intervals of the form (a, b0], where b0 is the largest element (ifany) of X.X does not have a largest element hence there are no basiselements of this type.

Sanjay Mishra The Order Topology

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Examples of order Topology V

Hence a basis for the order topology on X = {5} ∪ (0, 1) ∪ [4, 8) is

{(a, b) : (a ∈ [0, 1),& b ∈ (0, 1]) or (a ∈ [4, 8) & b ∈ (4, 8))}∪{(a, 1) ∪ [4, b) : a ∈ [0, 1), & b ∈ (4, 8)}∪{{5} ∪ (0, b) : b ∈ (0, 1]}∪

{{5} ∪ (0, 1) ∪ [4, b) : b ∈ (4, 8)} (1)

Another basis for this topology is

{(a, b) : (a ∈ (0, 1),& b ∈ (0, 1]) or (a ∈ [4, 8) & b ∈ (4, 8))}∪{(a, 1) ∪ [4, b) : a ∈ (0, 1), & b ∈ (4, 8]}∪{{5} ∪ (0, b) : b ∈ (0, 1)}∪

{{5} ∪ (0, 1) ∪ [4, b) : b ∈ (4, 8]} (2)

Sanjay Mishra The Order Topology

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Examples of order Topology VI

A basis for the order topology on Y = {5} ∪ [0, 1] ∪ [4, 8) is

{(a, b) : (a ∈ [0, 1), & b ∈ (0, 1]) or (a ∈ [4, 8) & b ∈ (4, 8))}∪{[4, b) : b ∈ (4, 8]}∪{[0, b) : b ∈ (0, 1]}∪{(a, 1] : a ∈ [0, 1)}∪{[0, 1]}∪{{5}}

(3)

Sanjay Mishra The Order Topology