# General Topology — Part 3

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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.

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}

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

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

Furthermore, the following 2 statements are equivalent:

1. A point x ∈ X is a **Limit Point** of A.

2. Every **Neighbourhood** of x contains a point of A different from x.

**Proof-**

1 ⇒ 2

∀N ∈ N(x)

⇒ ∃o ∈ τ s.t. x ∈ o ⊆ N

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

⇒ N\{x}∩A ≠ø **[ Since o ⊆ N ]**

2 ⇒ 1

∀o ∈ τ, x ∈ o

⇒ o\{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.

# 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:**

1. The **Closed Set** is a subset A ⊆ X where its **complement** is an **open set** in τ:

Ac = X\A ∈ τ

2. The **Closed Set** is a subset A ⊆ X which equals to its **Closure**:

A = Cl(A)

3. The **Closed Set** is a subset A ⊆ X which contains all its **Limit Point**:

A’ ⊆ A

**Proof-**

A is closed

⇒ A is the smallest closed set containing A

⇒ Cl(A) = A **[ By closure definition CL2 ]**

Cl(A) = A

⇒ A = A∪A’ **[ By closure definition CL4 ]**

⇒ A’ ⊆ A

A’ ⊆ A

⇒ A∪A’= A

⇒ A = Cl(A) **[ By closure definition CL4 ]**

⇒ A is closed **[ By closure definition CL2, Cl(A) is closed ]**

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].

# 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**.