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.
