General Topology — Part 6 (Hausdorff Space)

8 min readNov 9, 2023

Hausdorff Space and Compact Set

Hausdorff Space is a topological space (X, τ) in which for every p, q ∈ X, p ≠ q, there exist open set U and V such that p ∈ U, q ∈ V and U ∩ V = ∅.

As described in previous article (General Topology-Part5), Hausdorff Space has the following properties:

HD1. The compact set in the Hausdorff Space is closed
HD2. Rn is a Hausdorff Space

The Hausdorff-ness property can actually be extended to compact set:

HD3. Let X be the Hausdorff Space. For any p ∈ X and compact set A ⊆ X and A does not contain p, there exists open set U and V where U ∩ V = ∅ and p ∈ U and A ⊆ V.
Proof-
∀ p ∈ X
⇒ ∀ q ∈ A, there exist Uq and Vq s.t. p ∈ Uq and q ∈ Vq and Uq∩Vq= ∅ [ By definition of Hausdorff Space and p ∉ A, p ≠ q ]
⇒ ∪Vq ⊇ A and p ∈ Uq and Uq∩Vq= ∅
⇒ {Vq} is an open cover of A and {Uq} is a collection of neighbourhood of p
⇒ {V[i]} is a finite subcover of A and {U[i]} is a collection of neighbourhood of p [ Since A is compact. I use V[i] to denote V[0], V[1].., V[n] ]
⇒ ∪V[i] ⊇ A and p ∈ ∩U[i] is open [ By topology axiom T3, see Part 1 ]
⇒ There exists open set U=∩U[i] and V=∪V[i] s.t. p ∈ U and A ⊆ V and U ∩ V = ∩U[i] ∩ ∪V[j] = ∪(∩U[i] ∩ V[j]) = ∅ [ since U[i] ∩ V[i] = ∅ ]

HD4. Let X be the Hausdorff Space. For any disjoint compact set A, B ⊆ X, there exists open set U and V where U ∩ V = ∅ and A ⊆ U and B ⊆ V.
Proof-
∀ p ∈ A
⇒ There exist open set Up and Vp s.t. p ∈ Up and B ⊆ Vp and Up ∩ Vp= ∅ [By HD3 where p ∉ B and B is compact ]
⇒ {Up} is an open cover of A and Vp ⊇ B and Up∩Vp= ∅
⇒ {U[i]} is a finite subcover of A and V[i] ⊇ B [ Since A is compact ]
⇒ ∪U[i] ⊇ A and ∩V[i] is open and ∩V[i] ⊇ B [ By topology axiom T3, see Part 1 ]
⇒ There exists open set U=∪U[i] and V=∩V[i] s.t. A ⊆ U and B ⊆ V and
U ∩ V =∪U[i] ∩ ∩V[j] = ∪(U[i] ∩ (∩V[j])) = ∅ [ since U[i] ∩ V[i] = ∅ ]

When a compact set is contained by a union of 2 open sets, we can decompose this compact set into 2 smaller compact sets. Each open set can again contain the smaller compact set.

HD5. Let X be the Hausdorff Space. For a compact set K ⊆ X, and open sets U1,U2 ⊆ X where K ⊆ U1 ∪ U2, there exists compact sets K1, K2 s.t. K=K1∪K2 where K1⊆U1, K2⊆U2
Proof-
Let S1 = K\U1 = K ∩ (X\U1)
⇒ S1 is closed and S1 ⊆ K [ By HD1, K is closed and U1 is open ]
⇒ S1 is compact [ By CP1 ]
Let S2 = K\U2
⇒ S2 is compact [ By similar argument ]

S1∩S2 = K\U1 ∩ K\U2 = K\(U1 ∪ U2) ⊆ K\K = ∅
⇒ There exist open set V1, V2 such that S1 ⊆ V1, S2 ⊆ V2, V1 ∩ V2 = ∅ [ By HD4 ]

Let K1 = K\V1
⇒ K1 ⊆ K\S1 = K\(K\U1) = K ∩ U1 ⊆ U1
Let K2 = K\V2
⇒ K2 ⊆ K\S2 = K\(K\U2) = K ∩ U2 ⊆ U2
⇒ K1∪K2 = (K\V1)∪(K\V2) = K\(V1∩V2) = K\∅ = K

Compact Hausdorff Space

Compact Hausdorff Space is a topological space (X, τ) which is both Compact and Hausdorff.

Recall from previous article (General Topology-Part4), the Normal Space is defined as the topological space (X, τ) that for every two disjoint closed subsets A, B ⊆ X having disjoint open neighbourhoods:
∀A,B⊆ X, Ac, Bc ∈ τ, A⋂B=ø ⇒∃o1, o2 ∈ τ s.t. A ⊆ o1 and B ⊆ o2 and o1⋂o2=ø

Clearly, when a topological space is Compact and Hausdorff, then it is a Normal Space.

CH1. A Compact Hausdorff Space is a Normal Space.
Proof-
For every disjoint closed set A,B in Compact Hausdorff Space
⇒ A,B are disjoint compact set [ By CP1, closed set in compact space is compact ]
⇒ There exists open set U, V s.t. U⋂V = ø, A ⊆ U, B ⊆ V [ By HD4 ]
⇒ Compact Hausdorff Space is a Normal Space

Locally Compact Hausdroff Space

A less restrictive class of compact space is the locally compact space where each small portion of the space looks like a small portion of a compact space. The formal definition of Locally Compact Space is as follows:

Locally Compact Space is a topological space (X, τ) in which for every p∈X, p has a compact neighbourhood. i.e. there exists a compact an open set U and a compact set K, such that p ∈ U ⊆ K with U,K ⊆X.

Locally compact space arises naturally in various applications. For example, locally compact Hausdorff spaces are commonly encountered in analysis, functional analysis, and differential geometry. Many familiar spaces, such as Euclidean spaces Rn, are locally compact.

Locally Compact Hausdorff Space is a topological space (X, τ) which is Locally Compact and Hausdorff. Clearly, the Compact Hausdorff Space is a Locally Compact Hausdorff Space.

CH2. A Compact [Hausdorff] Space is a Locally Compact [Hausdorff] Space.
Proof-
Let (X, τ) be the Compact [Hausdorff] Space, for every p∈X, p has the compact neighbourhood X.
⇒ Compact [Hausdorff] Space is a Locally Compact [Hausdorff] Space

One-point compactification

Actually, the locally compact space can be extended to a compact space by the one point compactification process. The idea is to add a single new point “∞” to the original space and constructs a new collection of open sets based on the original open sets.

Let X be the Locally Compact Space (X, τ) and X* be the One-point compactification (X*, τ*) of X:

X* = X ∪ {∞}, τ* = τ ∪ { {∞} ∪ X\k | k is a closed compact set in X}

Note that ø is the closed and compact set, so k exists. Let’s prove that (X*,τ*) is a Topological Space and Compact:

Proof-
T1:
ø ∈ τ ⇒ ø ∈ τ*
ø is closed and compact ⇒ X* = {∞} ∪ X\ø ∈ τ*

T2:
∀U,V ∈ τ*
Case 1:
U,V ∈ τ
⇒ U∪V ∈ τ
Case 2:
U∈τ, V={∞}∪X\k where k is a closed compact
U∪V
= X\Uc ∪ {∞}∪X\k
= {∞} ∪ X\Uc ∪ X\k
= {∞} ∪ X\(Uc ∩ k) ∈ τ*
[ Uc is closed and k is closed ⇒ Uc ∩ k is closed
Uc ∩ k ⊆ k ⇒ Uc ∩ k is compact by CP1 ]
Case 3:
U={∞}∪X\k1, V={∞}∪X\k2 where k1 and k2 are closed compact
U∪V
= {∞}∪X\k1 ∪ {∞}∪X\k2
= {∞}∪X\k1∪X\k2
= {∞}∪X\(k1∩k2) ∈ τ*
[ k1 and k2 are closed ⇒ k1∩k2 is closed
k1∩k2 ⊆ k1 ⇒ k1∩k2 is compact by CP1 ]

T3:
∀U,V ∈ τ*
Case 1:
U,V ∈ τ
⇒ U∩V ∈ τ
Case 2:
U∈τ, V={∞}∪X\k where k is a closed compact
U∩V
= X\Uc ∩ ({∞}∪X\k)
= (X\Uc ∩ X\k) ∪ (X\Uc ∩ {∞})
= (X\Uc ∩ X\k) ∪ ø
= X\(Uc ∪ k) ∈ τ
[ Uc is closed and k is closed ⇒ Uc ∩ k is closed ]
Case 3:
U={∞}∪X\k1, V={∞}∪X\k2 where k1 and k2 are closed compact
U∩V
= ({∞}∪X\k1) ∩ ({∞}∪X\k2)
= {∞}∪(X\k1 ∩ X\k2)
= {∞}∪X\(k1∪k2) ∈ τ*
[ k1 and k2 are closed ⇒ k1∪k2 is closed
k1 and k2 are compact ⇒ k1∪k2 is compact by CP2 ]

∴ (X*, τ*) is a Topological Space

For any open cover {Uα} of X*
⇒ ∃ U[j] ∈ {Uα} where U[j]={∞}∪X\k [ Since ∞ ∈ X*, at least 1 Uα contains ∞ ]
⇒ {Uα | α≠j} ∪ {U[j]} is an open cover of X*
⇒ {Uα | α≠j} ∪ {U[j]} is an open cover of k [ k ⊆ X* ]
⇒ {Uα | α≠j} is an open cover of k [ U[j] ∩ k=ø ]
⇒ {U[i] | i≠j} is a finite subcover of k [ k is compact ]
⇒ {U[i] | i≠j} ∪ {Uj} is a finite subcover of k∪U[j]=k∪{∞}∪X\k={∞}∪X=X*
⇒ (X*, τ*) is compact

Consider the Locally Compact Hausdorff Space again. Can it be extended to a Compact Hausdorff Space by the one point compactification process? Yes, it can.

Let X be the Locally Compact Hausdorff Space (X, τ) and X* be the One-point compactification (X*, τ*) of X:

X* = X ∪ {∞}, τ* = τ ∪ { {∞} ∪ X\k | k is a closed compact set in X}

Let’s prove that (X*, τ*) is a Hausdroff Space:

Proof-
∀ p, q ∈ X*, p ≠ q,
Case 1:
p, q ∈ X
⇒ there exist open set U⊆X and V⊆X such that p ∈ U, q ∈ V, U ∩ V = ∅
[ X is Hausdorff ]
Case 2:
p ∈ X, q = ∞
⇒ there exist compact set k ⊆ X such that p ∈ k ⊆ X [ X is locally compact ]
⇒ there exist open set U ⊆ k, V = {∞} ∪ X\k s.t. p ∈ U and q ∈ V and
U ∩ V = k ∩ ({∞} ∪ X\k) = k ∩ X\k = ∅
[ k is compact and X is locally compact ]

Therefore, Locally Compact Hausdorff Space can be extended to the Normal Space by the one point compactification process.

Conclusion

In this post, we have studied the properties of compact set under Hausdroff Space. We find that Compact Hausdorff Space is a Normal Space. The definition of Locally Compact Space is introduced. By one-point compactification process, the Locally Compact [Hausdorff] Space can be extended to a Compact [Hausdorff] Space and the former one will be a subspace of the latter one. Sometimes, it is easier to prove the theorem in the extended space and then pass it back to its subspace, which is the original space we want to study. I will show you this interesting technique in next article.