Mathematics Analysis

1. EDMT 501: ANALYSIS
Presenter
Nabin Marahatta
Elements of Point Set Topology

2. Outlines
Countable & Uncountable Sets
Neighborhood
Deleted Epsilon Neighborhood
Interior Point & Interior
Exterior Point & Exterior
Boundary Point & Boundary
Adherent Points & Its Types
Open & Closed Sets
Bounded Sets
Bolzano-Weierstrass Theorem

3. Countable & Uncountable Sets
A set S is said to be denumerable or enumerable or countably infinite if and only if
there is a bijection
 𝑓: ℕ → 𝑆
i.e. if the set S is equivalent to the set ℕ = {1, 2, 3, … … … … … } then the set S is said
to be denumerable or enumerable or countably infinite. The cardinal number of
denumerable or enumerable or countably infinite is denoted by ℵ0.
The set S is said to be countable if and only if S is finite or denumerable.
The set S is said to be uncountable if S is neither finite nor denumerable.

4. Examples
𝑆 = 1, 2, 3 is a
countable set.
The set of positive
integers is countable set.
The set of rational
number is countable set.
The set of irrational
number is uncountable
set.
The set of real number is
uncountable set.

5. Neighborhood
Let 𝐴 be a subset of ℝ. Then set 𝐴 is called the neighborhood of a point 𝑐, if there
exist an open 1-ball 𝐵(𝑐; 𝑟) with centre c and radius r such that
𝐵 𝑐; 𝑟 ⊆ 𝐴
i.e. any set containing an open ball with centre 𝑐 is called a neighborhood of 𝑐.
Here, open 1-ball 𝐵 𝑐; 𝑟 = (𝑐 − 𝑟, 𝑐 + 𝑟) is entirely contained in the open interval
(𝑎, 𝑏). Therefore, 𝑎, 𝑏 is the neighborhood of each of its point.
𝑎 𝑏
𝑐 − 𝑟 𝑐 + 𝑟
𝑐
𝐴

6. Deleted
𝜺 −neighborhood
Let c 𝜖 ℝ and 𝜺 > 0. Then the set,
𝑁′𝜺 = 𝐵 𝑐; 𝜺 − 𝑐 = {𝑥: 0 < 𝑥 − 𝑐 < 𝜺}
is called a deleted 𝜺-neighborhood of c. i.e. an open
interval (𝑐 − 𝜺, 𝑐 + 𝜺) from which the number c
itself has been excluded or deleted is called a deleted
𝜺-neighborhood of c.

7. Examples
Every open 1-ball 𝐵(𝑐; 𝑟) is a neighborhood of its
centre.
The set of real numbers ℝ is neighborhood of each of its
points.
The open interval (𝑎, 𝑏) is neighborhood of each of its
points.
The closed interval [𝑎, 𝑏] is the neighborhood of each
point of (𝑎, 𝑏) but is not neighborhood of end-points a
and b.
The set of rational number ℚ is not the neighborhood of
any of its point.

8. Interior Point & Interior
Let 𝑆 be a subset of ℝ. Then the point 𝑐𝜖𝑆 is called the interior point of 𝑆, if there
exist an open 1-ball 𝐵(𝑐; 𝑟) such that
𝐵(𝑐; 𝑟) ⊆ 𝑆
i.e. a point c is said to be interior point of S ⊆ ℝ if it has a neighborhood 𝑈 lying
entirely inside 𝑆.
The set of all the interior point of a set 𝑆 is called the interior of the set 𝑆 and is
denoted by int. 𝑆.
𝑎 𝑏
𝑺
𝑰𝒏𝒕. 𝑺 = (𝒂, 𝒃)

9. Exterior Point & Exterior
Let 𝑆 be a subset of ℝ. Then the point 𝑐𝜖ℝ is called the exterior point of 𝑆, if there exist
𝑟 > 0 such that the open ball 𝐵(𝑐; 𝑟) entirely lies in 𝑆𝑐
= ℝ − 𝑆.
𝐵(𝑐; 𝑟) ⊆ 𝑆
i.e. a point c is said to be exterior point of S ⊆ ℝ if it has a neighborhood 𝑈 lying
entirely outside 𝑆.
The set of all the exterior point of a set 𝑆 is called the exterior of the set 𝑆 and is
denoted by ext (𝑆).
𝑎 𝑏
𝑺
𝑬𝒙𝒕. 𝑺 = −∞, 𝒂 ∪ (𝒃, ∞)

10. Boundary Point & Boundary
The point which is neither interior to 𝑆 nor exterior to S are called boundary
point.
The set of boundary point is called boundary. It is denoted by 𝑏𝑑(𝑆) or 𝛿𝑆.
𝑎 𝑏
𝑺
𝛿𝑆 = {𝒂, 𝒃}

11. Examples
Every points of open interval (𝑎, 𝑏) are its interior point. 𝑎 and 𝑏 are the boundary points of
(𝑎, 𝑏)
Let, S = {1, 2, 3} then int (S) = ∅, Ext. (S) = −∞, 1 ∪ (1,2) ∪ (2,3) ∪ (3, ∞) and 𝛿𝑆 = S.
Let, S = ℕ then int (S) = ∅, Ext. (S) = ℝ − ℕ and 𝛿𝑆 = ℕ.

12. Adherent Point
Let 𝑆 be a subset of ℝ. Then the point 𝑐𝜖ℝ is
called an adherent point of 𝑆, if every open
interval 𝐵(𝑐; 𝑟) of length 2𝑟 and centre at
𝑐 contains at least one point of 𝑆, i.e.
∀𝑟 > 0: 𝐵(𝑐; 𝑟) ∩ 𝑆 ≠ ∅
The set of all adherent points of 𝑆 is called
closure of 𝑆 and is denoted by ҧ
𝑆.

13. Types of Adherent Points
Adherent
point
Limit
point
Let 𝑆 be a subset of ℝ. Then the point 𝑐𝜖ℝ is called the limit
point of 𝑆, if every open interval 𝐵(𝑐; 𝑟) of length 2𝑟 and
centre at 𝑐 contains at least one point of 𝑆 other than c, i.e.
∀𝑟 > 0: [𝐵 𝑐; 𝑟 − 𝑐 ] ∩ 𝑆 ≠ ∅
The set of all limit points of 𝑆 is called the derived set of 𝑆 and
is denoted by 𝑆′.
Isolated
point
A point 𝑡𝜖𝑆 is called isolated point of 𝑆 if there exists a
neighborhood 𝑈 of 𝑡 such that 𝑈 ∩ 𝑆 = 𝑡 , i.e. the adherent
point which is not a limit point is called isolated point of 𝑆.

14. Examples:
❖ Limit point of a set S may or may not belong to S.
❖ Let, S = {1, 2, 3, 4}
Here, S is finite set. So, it has no limit point, i.e. S’ = ∅
❖ The set of natural number has no limit points, and every point of natural
number are boundary point.
❖ Every isolated point of S is a boundary point of S.
❖ The set of positive integer (ℤ+) has no limit points, and every point of ℤ+
are boundary points.

15. Open, Closed and Clopen Sets
A subset 𝑆 in ℝ is said to be open in ℝ if every point of 𝑆 is an interior point of 𝑆.
i.e. a subset 𝑆 of ℝ is said to be open set if it contains an open ball about each of its
points. That is 𝑆 ⊆ ℝ is open in ℝ if
∀ 𝑐 𝜖 ∃ 𝑟 > 0: 𝐵 𝑐; 𝑟 ⊆ 𝑆.
A subset 𝑆 in ℝ is said to be closed in ℝ if its complement 𝑆𝑐 = ℝ − 𝑆 is open in ℝ.
A set which is both open and closed is called a clopen set.

16. Examples
❖ Every non-empty finite set is closed.
❖ ℝ and ∅ are clopen sets.
❖ Every open interval is open set.
❖ [0, ∞) is closed set.

17. Bounded Sets
The subset 𝑆 in ℝ is said to be bounded above if there exist a real number 𝑀 such that
every member of 𝑆 is less than or equal to M, i.e. 𝑥 ≤ 𝑀 ∀ 𝑥 𝜖 𝑆.
The subset 𝑆 in ℝ is said to be bounded below if there exist a real number 𝑚 such that
every member of 𝑆 is greater than or equal to m, i.e. 𝑥 ≥ 𝑚 ∀ 𝑥 𝜖 𝑆.
The subset 𝑆 in ℝ is said to be bounded set if it is bounded below as well as bounded
above.
Examples:
[𝑎, 𝑏] and (𝑎, 𝑏) are bounded set.
𝑎, ∞ unbounded set.

18. Bolzano-Weierstrass Theorem
Statement: Every bounded infinite set in ℝ has a limit point in ℝ.
Proof:
Let 𝑆 be an infinite bounded set in ℝ.
By completeness axiom,
There exists supremum and infimum of 𝑆.
So, let Inf. S = m and Sup. S = M
𝑚 𝑀
−∞ ∞
𝑺

19. Bolzano-Weierstrass Theorem
Let, 𝐻 = {𝑥: 𝑥 𝜖 ℝ such that 𝑥 > infinite many elements of S} and 𝑀 𝜖 𝐻.
Here, m is one of the lower bound of H. So, by completeness axiom, H has infimum.
Let, inf. H= p
∴ 𝑝 ≤ 𝑥 ∀ 𝑥 𝜖 𝐻
𝑚 𝑀
−∞ ∞
𝑺
𝑯

20. Bolzano-Weierstrass Theorem
For arbitrary real number r such that 𝑟 > 0.
𝑝 ≤ 𝑥 < 𝑝 + 𝑟
𝑝 + 𝑟 will be greater than infinite elements of S.
Since, 𝑝 is infimum of 𝐻, (𝑝 − 𝑟) ∉ 𝐻 and (𝑝 − 𝑟) ∈ 𝑆
So, (𝑝 − 𝑟, 𝑝 + 𝑟) contains infinite elements of S other than p.
i.e. 𝐵 𝑝; 𝑟 − 𝑝 ∩ 𝑆 ≠ ∅
Therefore, p is a limit point of S.
Hence, every bounded infinite set in ℝ has a limit point in ℝ.

21. Examples:
a. Let, S = [2, 5] is an infinite bounded set.
Here, 𝐵 4, 0.1 − 4 ∩ 4, 0.1 ≠ ∅
So, , S = [2, 5] has limit points.
b. Let, S = (-3, 6) is an infinite bounded set.
Here, 𝐵 0, 1 − 0 ∩ −3, 6 ≠ ∅
So, , S = (-3, 6) has limit points.

22. Conclusion
❑ If S is finite or denumerable then S is called countable otherwise S is called
uncountable.
❑ Any set containing an open ball with centre 𝑐 is called a neighborhood of 𝑐 and the
set of neighborhood without centre is called deleted 𝜺-neighborhood of c.
❑ Point c is the interior point of 𝑆, if there exist an open 1-ball 𝐵(𝑐; 𝑟) in S.
❑ Point c is the exterior point of 𝑆, if there exist an open 1-ball 𝐵(𝑐; 𝑟) in ℝ −S.
❑ If every points of set S are interior points, then the set is open otherwise the set is
closed.
❑ Every bounded infinite set in ℝ has a limit point in ℝ.