General Topology — Part 3 (Dense Set)

Simon Kwan

9 min readDec 25, 2021

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

General Topology — Part 1

What is General Topology ?

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General Topology — Part 2

ε-δ Definition of Continuous Function

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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.