# General Topology — Part 3

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]
∴ 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.

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MSc Computer Science, BSc Math, BSc Mechanical Engineering