# General Topology — Part 3 (Dense Set)

In previous articles, I have discussed the definition of a topological space and continuous function.

In this article, I will discuss some essential topological concepts that are closely related: **Neighbourhood**, **Limit Point**, **Closed Set,** **Closure **and** Dense Set**. Let’s start our journey…

# Neighbourhood and Limit Point

Let (X, τ) be the topological space for set X.

The **Neighbourhood** N of a point x ∈ X is defined as the subset of X that includes an open set o containing x:

∃o ∈ τ s.t. x ∈ o ⊆ N

Since the open set o is a subset of X and includes itself, we can see that:

Every **Open Set** containing x is also a **Neighbourhood** of x. This set is called **Open Neighbourhood** of x.

Here is the definition of Open Set in terms of Neighbourhood:

NB1. A is an Open Set iff A is a neighbourhood of all its element**Proof-**

(only if)

∀ x∈A,

⇒ A is a open neighbourhood of x

⇒ A is a neighbourhood of x

(if)

∀ x∈A,

⇒ ∃o ∈ τ s.t. x∈o⊆A

⇒ ∃o ∈ τ s.t. {x}⊆o and o⊆A

⇒ A ⊆ Uo and Uo ⊆ A **[ Since U{x}=A ]**

⇒ A = Uo

⇒ A is open **[ By T2 ]**

The **Neighbourhood System** N(x) of a point x ∈ X is the collection of all neighbourhoods of the point x:

N(x) ={N | N⊆ X, ∃o ∈ τ s.t. x ∈ o ⊆ N}

NB2. Any finite intersection of neigbourhoods of x ∈ X is also a neigbourhood of x**Proof-**

V[1], V[2] ∈ N(x)

⇒ ∃ox1, ox2 ∈ τ s.t. ox1 ⊆ V[1] and ox1 ⊆ V[2]

⇒ ∃ox=ox1∩ox2 ∈ τ s.t. ox ⊆ V[1]∩V[2] ⊆ X

⇒ V[1]∩V[2] ∈ N(x)

The **Limit Point**** of a set** A ⊆ X is a point x ∈ X such that every open set ox around x, ox contains a point of A different from x:

∀ox ∈ τ, ox\{x}∩A ≠ø **[Note that x can be a member of set A and x∈ox ]**

Besides using open set, the limit point can be defined in terms of neighbourhood:

LP1. A point x ∈ X is a **Limit Point** of a set A ⊆ X iff every **Neighbourhood** of x contains a point of A different from x.

**Proof-**

(only if)

∀N ∈ N(x)

⇒ ∃ox ∈ τ s.t. x ∈ ox ⊆ N

⇒ ox\{x}∩A ≠ø **[ Since x is a limit point and definition of limit point]**

⇒ N\{x}∩A ≠ø **[ Since ox ⊆ N ]**

(if)

∀ox ∈ τ

⇒ ox\{x}∩A ≠ø **[ Since open set is also a neighbourhood ]**

# Closed Set and Open Set

The **Closed Set A **is a subset of X where its complement Ac is an open set in τ:

Ac = X\A ∈ τ

Similar to those topological conditions satisfied by **Open Set**, **Closed Set** has the following 3 properties:

CS1. ø, X are both closed sets

CS2. any finite union of closed sets is a closed set

CS3. any (finite or infinite) intersection of closed sets is a closed set

**Proof-**

Let A[i] be the closed set in τ

CS1. X=X\ø, ø=X\X,

⇒ X, ø are both closed sets under τ **[By topology axiom T1]**

CS2. A[i]∪A[j] = (X\Ac[i])∪(X\Ac[j]) = X\(Ac[i]∩Ac[j]),

Ac[i], Ac[j]∈τ ⇒ Ac[i]∩Ac[j]∈τ **[ By topology axiom T3 ]** ⇒ A[i]∪A[j] is a closed set

CS3. ∩A[i] = ∩(X\Ac[i]) = X\(∪Ac[i]),

Ac[i]∈τ ⇒ ∪Ac[i]∈τ **[ By topology axiom T2 ]** ⇒ ∩A[i] is a closed set

Note that both ø and X are open and closed at the same time.

# Closed Set in Subspace Topology

Let Y be a subspace of X and for subset A ⊆ Y ⊆ X:

CS4. A is closed in Y iff A = Y∩W for some W closed in X **Proof-**

(only if)

A is closed in Y

⇒ Y\A is open

⇒ Y\A = Y∩O for some O is open set in X

⇒ A=Y\(Y\A)=Y\(Y∩O)=Y\O=Y∩(X\O)=Y∩W for some W=X\O closed in X

(if)

A = Y∩W for some W closed in X

⇒ Y\A=Y\(Y∩W)=Y\W=Y∩(X\W)=Y∩O where O=X\W is a open set in X

⇒ Y\A is open in Y

⇒ A is closed in Y

# Closure of a Set

The **Closure** Cl(A) of a subset A ⊆ X is the intersection of all **Closed Sets** containing A.

**Closure** of a set can also be defined in terms of **Neighbourhood **and** Limit Point.** Denote the collection of all **Limit Points of set** A as A’. Let’s prove that the following equivalent definitions of **Closure**:

CL1. Cl(A) = ∩{c|A ⊆ c ⊆ X, c is a closed set}

CL2. Cl(A) is the smallest Closed Set containing A

CL3. Cl(A) = {x|x∈X, ∀N ∈ N(x), N∩A ≠ø}

CL4. Cl(A) = A∪A’

**Proof-**

Let T = {c|A ⊆ c ⊆ X, c is a closed set}

CL1 ⇒ CL2

Cl(A) = ∩c[i] where c[i]∈T

⇒ Cl(A) is closed **[ By closed set property CS3 ]**

∀c[i] ∈ T

⇒ A ⊆ c[i]

⇒ A ⊆ ∩c[i]

∀c[i] ∈ T

⇒ ∩c[i] ⊆ c[i]

∴ Cl(A) is the smallest Closed Set containing A

CL2 ⇒ CL3

∀x ∈ Cl(A)

Case 1: x ∈ A

⇒ ∀N ∈ N(x)

⇒ x ∈ N ⇒ x ∈ N∩A ⇒ N∩A ≠ø

Case 2: x ∈ X\A

Suppose ∃N ∈ N(x) s.t. N∩A=ø

⇒ x ∈ N and N ⊆ X\A

⇒ ∃o ∈ τ s.t. x ∈ o ⊆ N ⊆ X\A **[By definition of neighbourhood]**

⇒ o is an open set not containing A

⇒ o ∪ X\Cl(A) ∈ τ is an open set not containing A **[ By topology axiom T2] **

⇒ o ⊆ X\Cl(A) **[ By X\Cl(A) is the largest Open Set not containing A]**

⇒ x ∈ X\Cl(A) **[By x ∈ o]**

⇒ contradict x ∈ Cl(A)

∴ Cl(A) ⊆ {x|x∈X, ∀N ∈ N(x), N∩A ≠ø}

CL3 ⇒ CL4

∀x ∈ Cl(A)

Case 1: x ∈ A

⇒ x ∈ A∪A’

Case 2: x ∈ X\A

⇒ ∀N ∈ N(x), N∩A ≠ø

⇒ ∀N ∈ N(x), N\{x}∩A ≠ø **[ By N∩A = ((N\{x}∪{x})∩A = (N\{x}∩A)∪({x})∩A) and x ∉ A ]**

⇒ x ∈ A’ ⇒ x ∈ A∪A’

∴ Cl(A) ⊆ A∪A’

CL4 ⇒ CL1

∀x ∈ Cl(A)

Case 1: x ∈ A

⇒ x ∈ ∩c[i] where c[i] ∈ T **[ By A ⊆ c[i] ∀c[i] ∈ T]**

Case 2: x ∈ A’

Suppose x ∉ ∩c[i] where c[i] ∈ T

⇒ x ∈ X\∩c[i] **[ By CS3, ∩c[i] is closed, X\∩c[i] is open ]**

⇒ x ∈ ∪(X\c[i])

⇒ ∃ X\c[j] ∈ τ s.t. x ∈ X\c[j] **[ By c[j] is closed and X\c[j] is open ]**

⇒ (X\c[j])\{x} ∩ A ≠ø **[ X\c[j] is open and thus X\c[j] ∈ N(x) and then by definition of A’ ] **

⇒ (X\A)\{x} ∩ A ≠ø **[ By A ⊆ c[j] ]**

⇒ (X\A) ∩ A ≠ø

⇒ contradict (X\A) ∩ A =ø

∴ Cl(A) ⊆ ∩c[i] where c[i] ∈ T

Since we have proved CL1⇒CL2⇒CL3⇒CL4⇒CL1, they are equivalent definitions of **Closure.**

Applying the definitions of **Closure, **we find out 2 more additional definitions of a **Closed Set**. For A ⊆ X**:**

CL5. A is closed in X iff Ac = X\A ∈ τ

CL6. A is closed in X iff A = Cl(A)

CL7. A is closed in X iff A contains all **its** **Limit Point**: A’ ⊆ A

**Proof-**

CL6.

A is closed

⇔ A is the smallest closed set containing A

⇔ A = Cl(A) [ By closure definition CL2 ]

CL7.

A is closed

⇔ A = Cl(A) [ By closure definition CL6 ]

⇔ A = A∪A’ [ By closure definition CL4 ]

⇔ A’ ⊆ A

Take interval [0, 1] as an example for the Closed Set. It is a closed set since every point within this interval is a limit point, you can find a neighbourhood of this point that intersects [0, 1].

Below are the useful properties of **Closure **for any subset A, B ⊆ X:

CL8. A ⊆ B ⇒ Cl(A) ⊆ Cl(B)

CL9. A ⊆ Bc and B is open ⇒ Cl(A) ⊆ Bc

CL10. Cl(A∩B) ⊆ Cl(A) ∩ Cl(B)

CL11. Cl(A∪B) = Cl(A) ∪ Cl(B)**Proof-**

CL8.

A ⊆ B and B ⊆ Cl(B) **[ By CL4 ]**

⇒ A ⊆ Cl(B)

⇒ A ⊆ Cl(A) ⊆ Cl(B) **[ By CL2, Cl(B) is a closed set and Cl(A) is the smallest closed set containing A, so Cl(A) is contained by Cl(B) ]**

CL9.

A ⊆ Bc and B is open

⇒ Bc is closed

⇒ A ⊆ Cl(A) ⊆ Bc **[ By CL2, Cl(A) is the smallest closed set containing A ]**

CL10.

A∩B ⊆ A and A∩B ⊆ B

⇒ Cl(A∩B) ⊆ Cl(A) and Cl(A∩B) ⊆ Cl(B) **[ By CL8 ]**

⇒ Cl(A∩B) ⊆ Cl(A)∩Cl(B)

CL11.

A ⊆ A∪B and B ⊆ A∪B

⇒ Cl(A) ⊆ Cl(A∪B) and Cl(B) ⊆ Cl(A∪B) **[ By CL8 ]**

⇒ Cl(A)∪Cl(B) ⊆ Cl(A∪B)

A ⊆ Cl(A) and B ⊆ Cl(B) **[ By CL4 ]**

⇒ A∪B ⊆ Cl(A)∪Cl(B)

⇒ A∪B ⊆ Cl(A∪B) ⊆ Cl(A)∪Cl(B) **[ By CL2, Cl(A)∪Cl(B) is a closed set. By CL4, Cl(A∪B) is the smallest closed set containing A∪B ]**

# Closed Set and Limit of Sequence

The **Limit Point**** of a ****sequence** (z[k]) in X is a point x ∈ X where for every neighbourhood V∈N(x), there exists an integer n such that z[k] ∈ V when k **≥** n:

z[k] → x ⇔ ∀V∈N(x), ∃n∈** N** ∀k

**≥**n z[k]∈V

Consider the case when all sequence (z[k]) ⊂ A. What would happen when A contains all the limit point of (z[k]) ?

CS5. A is closed in X iff A contains all the Limit Point of (z[k]) ⊂ A**Proof-**

(only if)

Let z[k] → x ∈ X and (z[k]) ⊂ A and suppose x ∈ Ac ∈ τ

⇒ ∃n∈N ∀k≥n z[k]∈Ac **[ Since z[k] → x and Ac ∈ N(x) and then by NB1 ]**

⇒ contradict ∀k z[k]∈A

(if)

Suppose A is not closed

⇒ Ac is not open

⇒ ∃x ∈ Ac s.t. Ac ∉ N(x) **[ By negation of NB1 ]**

⇒ ∀ox∈τ ox∩(Ac)c≠ø **[ By negation of definition of neighbourhood ]**

⇒ ∀V∈N(x) ∃ox∈τ and ox⊆V s.t. V∩A ⊇ ox∩A≠ø

Consider indexed family {V[k]} = N(x) where k=1,2,..

Consider S[k]= V[1]∩V[2]∩..V[k]∩A

⇒ S[k]≠ø **[ By NB2, ∩V[k] ∈ N(x) and ∀V∈N(x) V∩A≠ø ]**

⇒ (z[k]) exists **[ By ****axiom of choice****, pick z[k]∈ S[k] for k=1,2.. ]**

⇒ ∀V[k]∈N(x), ∀j≥k z[j]∈V[k] **[ Since z[j] ∈ S[j]=V[1]∩V[2]∩..V[j]∩A ]**

⇒ z[k] → x and (z[k]) ⊂ A with x ∈ Ac

⇒ contradict x ∈ A **[ Since A contains all the Limit Point of (z[k]) ⊂ A ]**

# Closure of a set in Subspace Topology

Let Y be a subspace of X and for subset A ⊆ Y ⊆ X:

CL12. ClY(A) = Y ∩ ClX(A) **[ I use ClX(A) and ClY(A) to denote the closure of A in topology (X,τX) and (Y,τY) respectively ]**

**Proof-**

ClY(A) = ∩{c | c is a closed set in Y and A ⊆ c}

= ∩{Y∩d | d is a closed set in X and A ⊆ d} **[ By CS4, c=Y∩d ]**

= Y ∩ (∩{d | d is a closed set in X and A ⊆ d})

= Y ∩ ClX(A)

# Dense Set

The **Closure** of a subset A ⊆ X can be as smallest as subset A or as largest as set X. We have seen the former case in previous paragraph, how about the latter one? Topologist has given a name to this kind of set:

The **Dense Set A in X **is a subset A ⊆ X where its **Closure** is the whole set X:

Cl(A) = X

From the equivalent definitions of **Closure** Cl(A), we get the following conditions for the subset A being **Dense in X**:

DS1. Cl(A) = X

DS2. X is the smallest **Closed Set** containing A **[From CL2]**

DS3. ∀x∈X ∀N ∈ N(x), N∩A ≠ø **[From CL3]**

DS4. A∪A’ = X **[From CL4]**

Obviously, we can see that **X is Dense Set in X** since set X is the smallest **Closed Set** containing X.

# Conclusion

In this post, many essential topological concepts are introduced including Neighbourhood, Limit Point, Closed Set, Closure** **and** **Dense Set. Let’s recap what we have discussed:

**Closed Set** is a subset A ⊆ X which has open complement**, **contains all its Limit Point or equals to its Closure.

The **Closure** of a set A is the smallest Closed Set containing A, the set A together with all of its limit points, the intersection of all closed sets containing, or contains all point x with its every neighbourhood intersects A.

A set A is said to be **Dense **in set X when closure of A equals to X, the smallest closed Set containing A is X or every point of X either belongs to A or is a limit point of A.

In the next article, I will show you one application of topology theory for constructing continuous function — **Urysohn’s lemma**.

# Your feedback is highly appreciated and will help me to continue my articles.

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Reference

“https://en.wikipedia.org/wiki/Dense_set”

“https://en.wikipedia.org/wiki/Closure_(topology)”

“https://en.wikipedia.org/wiki/Closed_set”

“https://en.wikipedia.org/wiki/Neighbourhood_(mathematics)”

“https://en.wikipedia.org/wiki/Limit_point”

“https://en.wikipedia.org/wiki/Sequence”

“https://en.wikipedia.org/wiki/Limit_of_a_sequence”

“https://en.wikipedia.org/wiki/Axiom_of_choice”

“Topology (2nd edition) by James Munkres”