Syllabus (UNIT-I, C3T, SEM - II) Intervals, Limit points ...

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Intervals

Syllabus (UNIT-I, C3T, SEM - II)

Intervals, Limit points of a set, Isolated points, Open set, closed set, derived set, Illustrations of Bolzano Weierstrass theorem for sets, compact sets in R, Heine-Borel Theorem.

Intervals

Definition: An interval is a set of real numbers lying between two numbers called the extremities of the interval.

For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1 and all numbers in between.

Notation for Intervals:

The interval of numbers between a and b, including a and b, is often denoted [a, b]. The

two numbers are called the endpoints of the interval.

Including or excluding endpoints:

𝑶𝒑𝒆𝒏 ∶ (𝒂, 𝒃) = {𝒙 ∈ 𝑹 | 𝒂 < 𝑥 < 𝑏},

𝑳𝒆𝒇𝒕 𝑪𝒍𝒐𝒔𝒆𝒅, 𝒓𝒊𝒈𝒉𝒕 𝒐𝒑𝒆𝒏: [𝒂, 𝒃) = {𝒙 ∈ 𝑹 | 𝒂 ≤ 𝒙 < 𝑏},

𝑳𝒆𝒇𝒕 𝒐𝒑𝒆𝒏, 𝒓𝒊𝒈𝒉𝒕 𝒄𝒍𝒐𝒔𝒆𝒅: (𝒂, 𝒃] = {𝒙 ∈ 𝑹 | 𝒂 < 𝑥 ≤ 𝑏},

𝑪𝒍𝒐𝒔𝒆𝒅: [𝒂, 𝒃] = {𝒙 ∈ 𝑹 | 𝒂 ≤ 𝒙 ≤ 𝒃}.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Intervals

Each interval (a, a), [a, a), and (a, a] represents the empty set, whereas [a, a] denotes the set {a}.

Infinite endpoints

In some contexts, an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with −∞ and +∞.

In this interpretation, the notations [−∞, b] , (−∞, b] , [a, +∞] , and [a, +∞) are all meaningful and distinct. In particular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals. (0, +∞) is the set of positive real numbers also written ℝ+.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Neighbourhood, Interior Points, Open Sets

Neighbourhood

Definition:Let c ∈ 𝑅. A subset 𝑆 ⊂ 𝑅 is said to be a neighbourhood of c if

there exists an open interval (a,b) such that 𝑐 ∈ (𝑎, 𝑏) ⊂ 𝑆.

Results:

1. The union of two neighbourhoods of c is a neighbourhood of c.

2. The intersection of two neighbourhoods of c is a neighbourhood of

c.

3. The intersection of a finite number of neighbourhoods of c is a

neighbourhood of c.

4. The intersection of an infinite number of neighbourhoods of c is

may not be a neighbourhood of c.

For Example, for every n ∈ 𝑵, (-1/n, 1/n) is a neighbourhood of 0.

⋂ (−𝟏

𝒏,

𝟏

𝒏)∞

𝒏=𝟏 = {0} is not a neighbourhood of 0.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Neighbourhood, Interior Points, Open Sets

Interior Point: If S is a subset of a R, then x in S is an interior point of S if there exists a

neighbourhood N(x) of x such that 𝑁(𝑥) ⊂ 𝑆.

Interior: The interior of a set S is the set of all interior points of S. The interior of S is denoted int(S), Int(S) or So. The interior of a set has the following properties.

int(S) is an open subset of S.

int(S) is the union of all open sets contained in S.

int(S) is the largest open set contained in S.

A set S is open if and only if S = int(S).

int(int(S)) = int(S) (idempotence).

If S is a subset of T, then int(S) is a subset of int(T).

If A is an open set, then A is a subset of S if and only if A is a subset of int(S).

Examples:

In any space, the interior of the empty set is the empty set.

If X is the Euclidean space of real numbers, then int([0, 1]) = (0, 1).

The interior of the set of rational numbers is empty.

In any Euclidean space, the interior of any finite set is the empty set.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Neighbourhood, Interior Points, Open Sets

Open Set

Definition: A subset S of R is open in R if for each x ∈ S there is a neighborhood N(x) of

x such that N(X) ⊆ S. In otherwords a subset S of R is open in R if each point of S is an

interior point of S.

Example:

1. R is open.

2. (0, 1) is open.

3. (a, b) is open.

4. ∅ is open.

Results:

1. The union of a finite collection of open subsets in R is open.

2. The union of an arbitrary collection of open subsets in R is open.

3. The intersection of any finite collection of open sets in R is open.

4. The intersection of an arbitrary collection of open sets in R may not be open.

Example: 𝟏. ⋂ (− 𝟏

𝒏, 𝟏

𝒏)∞

𝒏=𝟏 = {𝟎}. This is not open set.

2. ⋂ (−𝒏, 𝒏)∞𝒏=𝟏 = (𝟎, 𝟏). This is an open set.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Limit Points, Closed Sets, Compact Sets

Limit Point: If S is a subset of R. A point x in R is a limit point (or cluster

point or accumulation point) of S if every neighbourhood of x contains at least one

point of S different from itself.

Isolated Point : If S is a subset of R. A point x in S is an isolated point of S if it is not a

limit point of S.

Examples:

1. Let 𝑺 = {1,𝟏

𝟐,

𝟏

𝟑, … … . . }. Every point of S is an isolated point of S. 0 is a limit

point of S.

2. Let S= Z. Then every point of Z is an isolated point of Z.

3. Let S= N. Then every point of N is an isolated point of N.

4. Let S= Q. Then every point of Q is an isolated point of Q.

5. Let S= R. Then every point of R is a limit point of R.

Bolzano Weierstrass Theorem: Every bounded infinite set of real numbers has at

least one limit point in R.

Proof: Refer to textbook.

The Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass.

Derived Set: Let S be a subset of R. The set of all limit points of S is said to be the

derived set of S and is denoted by S’.

Examples:

1. Let S be a finite Set. Then S’=Ф.

2. Let S=N. Then S’=Ф.

3. Let S=Z. Then S’=Ф.

4. Let S=Q. Then S’=Ф.

5. Let S=R. Then S’=R.

6. Let S=Ф. Then S’=Ф.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Limit Points, Closed Sets, Compact Sets

Closed Set: A closed set is a set whose complement is an open set. In otherwords a set

is closed if it contains all its limit points.

Examples:

1. R is closed.

2. [0, 1] is closed.

3. [a, b] is closed.

4. ∅ is closed.

5. Z is a closed set.

6. N is a closed set.

7. Q is not a closed set.

Results:

1. The union of a finite collection of closed subsets in R is closed.

2. The intersection of any finite collection of closed sets in R is closed.

3. The intersection of an arbitrary collection of closed sets in R is closed.

4. The union of an arbitrary collection of closed subsets in R may not be closed.

Example:

𝟏. ⋃ [− 𝟏

𝒏, 𝟏

𝒏]∞

𝒏=𝟏 = [−𝟏, 𝟏]. This is a closed set.

2. ⋃ [𝟏

𝒏, 𝟑 −

𝟏

𝒏]∞

𝒏=𝟏 = (𝟎, 𝟑). This is not a closed set.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Limit Points, Closed Sets, Compact Sets

Open Cover: Let S be a subset of R. An open cover of S is a collection G= {Gα} of open sets

in R whose union contains S; i.e.,𝑆 ⊆ ∪ 𝐺𝛼.

If G’ is a subcollection of sets from G such that the unions of the sets in G’ contains S,

then G’ is called a subcover of G.

If G′ consists of finitely many sets, then we call G′ a finite subcover of G.

Examples:

Suppose S = (0, 1] some coverings of S are:

1. G0 = {(−1, 2)}

2. G1 = {(1/n, 2) : n ∈ N}

3. G2 = {(n − 1, 2) : n ∈ N}

4. G3 = {(1/n, n) : n ∈ N}

5. G4 = {(−1/n, 1 + 1/n) : n ∈ N}.

Heine-Borel Theorem: Let S be a closed and bounded subset of R. Then every open

cover of S has a finite sub cover.

Compact Set: Let S be a subset of R. S is said to be compact set if every open cover G of

S has a finite subcover.

Examples:

1. [0,1] is compact.

2. (0,1) is not compact.

3. Q is not compact.

Note: Heine-Borel Theorem states that a closed and bounded subset of R is compact.

Converse of Heine-Borel Theorem: A compact subset of R is closed and bounded in

R.

Dr. Shreyasi Jana Assistant Professor, Dept. of Mathematics

Narajole Raj College

C3T(SEM II), Unit-I, Limit Points, Closed Sets, Compact Sets

Exercise

1. Prove that an interval is an open set.

2. Let 𝑺 = {1,𝟏

𝟐,

𝟏

𝟑, … … . . }. Show that 0 is a limit point of S.

3. Show that a finite set has no limit points.

4. Find the derived set of 𝑺 = {1

𝑛, 𝒏 ∈ 𝑵}.

5. Find the derived set of 𝑺 = {(−𝟏)𝒎 +1

𝑛, 𝒏 ∈ 𝑵}.

6. Examine if the set of 𝑺 = {𝟏

𝒎+

1

𝑛, 𝒎, 𝒏 ∈ 𝑵} is open or closed.

7. Prove that a subset S of R is closed if and only if S’⊂S.

8. Show that Q is not compact.

9. Show that (0,1) is not compact.

10. Show that R is not compact.