# Homeomorphisms of Compact Sets in Certain Hausdorff Spaces

Authors: Arthur D.  Grainger
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We construct a class of Hausdorff spaces (compact and noncompact) with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic. Also, it is shown that these spaces contain compact subsets that are infinite.


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## Body

## 1. Introduction

In this paper, we construct a class of Hausdorff spaces with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic (Theorem3.7).  Conditions are given for these spaces to be compact (Corollary 2.10).  Also, it is shown that these spaces contain compact subsets that are infinite (Corollary 2.10).This paper uses the Zermelo-Fraenkel axioms of set theory with the axiom of choice (see [1–3]). We let ω denote the finite ordinals (i.e., the natural numbers) and ℕ denotes the counting numbers (i.e., ℕ=ω∖{0}). Also, for a given set X, we denote the collection of all subsets of X  by 𝒫(X), and we denote the cardinality of X by |X|. In other words, |X| is the smallest ordinal number for which a bijection of |X| onto X exists.In this paper, we will only consider compact topologies that are Hausdorff.  A topologyτ  on a set  X  is compact if and only if 𝒜⊆τ  and  X⊆⋃𝒜  imply  X⊆⋃j=1nUj  for some  n∈ℕ  and  {U1,…,Un}⊆𝒜.  Therefore, compact topologies need not be Hausdorff.


## 2. A Class of Hausdorff Spaces

LetV, W, and x0  be sets such that W is infinite and the collection
(2.1){V,W,{x0}}
is pairwise disjoint. For example, let  V={(ν,0)∣ν∈ω},  W={(μ,1)∣μ∈2ω}, and x0=22ω.  Unless otherwise stated, we let
(2.2)X={x0}∪V∪W.Recall that for setY  and  G⊆Y, we have
(2.3)Y∖G={y∈Y∣y∉G}.Definition 2.1.
LetA be an infinite set. Define
(2.4)Fr(A)={F∈P(A)∣|A∖F|∈ω}.
We call  ℱr(A)  the Fréchet filter on A.Note thatA  being infinite  implies that ℱr(A)  is a filter (see [4, Definition  3.1, page 48]).Definition 2.2.
Consider the collectionℬ1⊆𝒫(X)  defined as follows:
(2.5)B1=P(X∖{x0})∪{F∪{x0}∣F∈Fr(W)}.Proposition 2.3.
The collectionℬ1  generates a Hausdorff topology τ on  X.Proof.
Clearly,ℬ1 is a basis for a topology τ  (see [5, Section  13]).
Letv,w∈X  such that  v≠w.  If  v,w∈X∖{x0},  then  {v},{w}∈ℬ1,  v∈{v},  w∈{w},  and
(2.6){v}∩{w}=∅.
If  v=x0,  then  either w∈V or  w∈W. Assume that  w∈V,  and let  F∈ℱr(W).  Since  V∩W=∅,  {w}∈𝒫(V)⊆ℬ1 and  F∪{x0}∈ℬ1,  we have
(2.7){w}∩[F∪{x0}]=∅.
Assume that   w∈W.  Note that  {w}∈ℬ1.  Also,  W∖{w}∈ℱr(W), which implies  [W∖{w}]∪{x0}∈ℬ1  and
(2.8)v=x0∈[W∖{w}]∪{x0}.
Observe that,
(2.9){w}∩([W∖{w}]∪{x0})=∅.
We infer thatτ  is  Hausdorff.Proposition 2.4.
IfA⊆X∖{x0},  then  A  is compact in (X,τ) if and only if  A  is a finite set.Proof.
Note that finite sets are compact in any topological space.  So, assume thatA  is an infinite, and let
(2.10)A={{a}∣a∈A}⊆P(X∖{x0})⊆τ,
which implies
(2.11)A=⋃A.
Let  𝒰⊆𝒜  be a nonempty, finite subcollection of 𝒜. Therefore, there exists {ai}i=1m⊆A,  for some m∈ℕ, such that
(2.12)U={{ai}}i=1m,
which implies
(2.13)⋃U={ai}i=1m.
If  A⊆⋃𝒰, then we would have  A⊆{ai}i=1m, contradicting A being an infinite set.  Consequently,  infinite subsets of  X∖{x0}  are not compact in the topological space  (X,τ).Corollary 2.5.
The setW is not compact in  (X,τ).Corollary 2.6.
The setV is compact in  (X,τ) if and only if  V is finite.Proposition 2.7.
LetA⊆X∖{x0}. The set  A∪{x0} is compact in (X,τ)  if and only if   A∩V  is a finite set.Proof.
The topologyτ  on  X  is generated by ℬ1 (see Proposition 2.3).
Assume thatA∩V  is an infinite set. Let F∈ℱr(W), and let  Q=F∪{x0}. Hence,
(2.14)x0∈Q,Q∈τ.
Let  𝒜A={{a}∣a∈A},  and let  𝒜={Q}∪𝒜A.  Note that  𝒜A⊆τ,   𝒜⊆τ,
(2.15)A=⋃AA,A∪{x0}⊆⋃A.
Suppose that  𝒰⊆𝒜A is a finite subcollection such that
(2.16)A∪{x0}⊆⋃U.
It can be assumed, without loss of generality, that  Q∈𝒰  and  𝒰={Ui}i=0m for  m+1=|𝒰|∈ℕ  such that  U0=Q. So, by definition of  𝒜, there exists  {ai}i=1m⊆A such that  Ui={ai}  for  i=1,...,m.  Note that Q=F∪{x0}⊆W, which implies
(2.17)Q∩(A∩V)=A∩(Q∩V)=∅.
So, expressions (2.16)  and  (2.17) would imply
(2.18)A∩V⊆⋃i=1mUi={a1,…,am},
contradicting A∩V  being an infinite set.  We infer that  A∪{x0}  is not compact in (X,τ).
Conversely, assume thatA∩V  is a finite set. Let  𝒜⊆τ such that
(2.19)A∪{x0}⊆⋃A.
Hence, there exists U∈𝒜 such that  x0∈U. Since x0∉E  for E∈𝒫(X∖{x0}), there exists F∈ℱr(W) such that
(2.20)x0∈F∪{x0}⊆U.
Assume that  A∩V={vi}i=1m  for some  m∈ω  [we define {vi}i=10=∅]. Observe that,
(2.21)A=A∩[V∪W]=([A∩F]∪[A∩W∖F])∪{vi}i=1m.
Since  W∖F  is finite,  we can assume that
(2.22)A∩W∖F={aq}q=1k,
for some  k∈ω (we define {aq}q=10=∅). Hence, there exists {Ui}i=1m⊆𝒜  and {Uq}q=1k⊆𝒜  such that
(2.23)vi∈Ui,aq∈Uq,
for  i=1,…,m  and  q=1,...,k  (again, we define ⋃i=10Ui=⋃q=10Uq=∅).  Consequently, from expressions (2.19),  (2.20),  (2.21), (2.22), and (2.23),  we have
(2.24)A∪{x0}⊆U∪[⋃i=1mUi]∪[⋃q=1kUq].
We infer   A∪{x0} is compact in (X,τ).Corollary 2.8.
The setW∪{x0} is compact in  (X,τ).Proof.
Note thatW⊆X∖{x0}. Also, W∩V=∅ implies that  W∩V is finite. Therefore, W∪{x0}  is compact in (X,τ) by Proposition 2.7.Corollary 2.9.
The setV∪{x0}  is compact in (X,τ) if and only if  V is finite.Proof.
Note thatV⊆X∖{x0} and V=V∩V. Therefore,  the corollary follows from Proposition 2.7.Corollary 2.10.
The topological space(X,τ) is compact if and only if  V is finite.Proof.
Observe thatX=(X∖{x0})∪{x0}  and  V=(X∖{x0})∩V. Therefore,  the corollary follows from Proposition 2.7.Proposition 2.11.
IfK⊆X  is an infinite, compact set (in (X,τ)), then  K=A∪{x0} for some infinite set A⊆X∖{x0}  such that  A∩V is a finite set.Proof.
Ifx0∉K, then we would have K⊆X∖{x0}, contradicting Proposition 2.4, since K is an infinite, compact set.  Hence,  K=(K∖{x0})∪{x0} and (K∖{x0})∩V is a finite set by Proposition 2.7, since (K∖{x0})∪{x0} is compact. Also, note that K being an infinite set implies K∖{x0}  is an infinite set.   Let  A=K∖{x0}.Notation 2.12.
LetZ be a nonempty set, and let  θ be a Hausdorff topology on Z. For z∈Z, we let 𝒩θ(z) denote the filter of θ-neighborhoods of z; that is, U∈𝒩θ(z)  if and only if  z∈O⊆U  for some O∈θ.Recall that  forA⊆X  and  x∈X,  x is an accumulation point of A if and only if  for  U∈𝒩τ(x), there exists a∈A  such that  a≠x  and  a∈U.Remark 2.13.
IfA⊆X  and  x∈X∖{x0}, then x is not an accumulation point of A.Indeed,x∈X∖{x0}  implies  {x}∈𝒩τ(x). So, if  a∈A such that  a≠x,  then  a∉{x}.Remark 2.14.
The elementx0 is an accumulation point of  X∖{x0}in(X,τ).Indeed, letU∈𝒩τ(x0).  Hence,  {x0}∪F⊆U  for some F∈ℱr(W).  Since W is infinite and x0∉W, we have that  F∖{x0}=F≠∅.  Let  y∈F.  Hence,  y∈X∖{x0}  and  y∈U.Consequently,X∖{x0} is not closed [in (X,τ)], which implies {x0} is not open; that is,  {x0}∉τ.  In fact, x0 is the only element of  X such that {x0}∉τ.


## 3. Homeomorphisms of Compact Sets in(X,τ)

The following proposition is obvious and is stated without proof.Proposition 3.1.
Letκ be an infinite cardinal. If  D and G are sets such that  x0∉D∪G and
(3.1)|D|=κ=|G|,
then there exists a map  ζ:D∪{x0}→G∪{x0}  such that  ζ(x0)=x0 and ζ is a bijection.Remark 3.2.
IfF∈ℱr(W) and Z⊆X is a finite set, then  F∖Z∈ℱr(W).Lemma 3.3.
LetJ and K  be nonempty subsets of  X∖{x0} such that  J∩V  and  K∩V  are finite sets. Let φ:J∪{x0}→K∪{x0}  be a bijection such that φ(x0)=x0. If F∈ℱr(W),  then
(3.2)(K∪{x0})∩(E∪{x0})⊆φ[(J∪{x0})∩(F∪{x0})]
for some  E∈ℱr(W).Proof.
LetF∈ℱr(W). Note that
(3.3)J∪{x0}=[(J∪{x0})∩(F∪{x0})]⋃(J∩(W∖F))⋃(J∩V).
Also, the sets J∩(W∖F) and  J∩V are finite.
Let(3.4)Z=φ[(J∩(W∖F))⋃(J∩V)].
Consequently,  Z⊆X∖{x0}   and Z is a finite set.
Let(3.5)E=F∖Z.
Therefore,  E∈ℱr(W) by Remark 3.2. Note that
(3.6)(F∪{x0})∖Z=E∪{x0}.
So,
(3.7)(K∪{x0})⋂(E∪{x0})=[(K∪{x0})⋂(F∪{x0})]∖Z.
Observe that,
(3.8)φ[J∪{x0}]=φ[[(J∪{x0})∩(F∪{x0})]⋃(J∩(W∖F))⋃(J∩V)]=φ[(J∪{x0})∩(F∪{x0})]⋃φ[(J∩(W∖F))⋃(J∩V)]=φ[(J∪{x0})∩(F∪{x0})]⋃Z.
Therefore,
(3.9)(φ[J∪{x0}])∖Z=(φ[(J∪{x0})∩(F∪{x0})]⋃Z)∖Z=(φ[(J∪{x0})∩(F∪{x0})])∖Z⊆φ[(J∪{x0})∩(F∪{x0})].
Consequently,
(3.10)(K∪{x0})∩(E∪{x0})=[(K∪{x0})∩(F∪{x0})]∖Z⊆(K∪{x0})∖Z=(φ[J∪{x0}])∖Z⊆φ[(J∪{x0})∩(F∪{x0})].Proposition 3.4.
Letτ be the Hausdorff topology on X generated by ℬ1. Let A and B be infinite subsets of  X∖{x0} such that  A∩V  and  B∩V are finite sets.  If  A and B have the same cardinality (i.e., |A|=|B|), then any bijection of A∪{x0} onto B∪{x0} that has  x0  as a fixed point is a homeomorphism.Proof.
Letζ:A∪{x0}→B∪{x0}  be a  bijection with ζ(x0)=x0. Note that the existence of  such a bijection is established by Proposition 3.1.  Let  ζ-1:B∪{x0}→A∪{x0}  denote the inverse map of ζ.
Letλ  be the topology on A∪{x0}  induced by τ, and let  ρ  be the topology induced on  B∪{x0}  by  τ.
LetU∈λ.  We will show that  ζ[U]∈ρ.
Letb∈ζ[U].  Either b=x0 or b∈B.Case 1.
Assume thatb=x0.  Hence,  x0=ζ-1(x0)=ζ-1(b)∈U.  So, there exists Q∈ℬ1 such that x0∈(A∪{x0})∩Q⊆U.  Since  x0∉D  for any  D∈𝒫(X∖{x0}),  we have that Q=F∪{x0} for some  F∈ℱr(W).  Therefore,
(3.11)(A∪{x0})∩Q=(A∪{x0})∩(F∪{x0}),
which implies
(3.12)x0∈(A∪{x0})∩(F∪{x0})⊆U,
which implies
(3.13)b=ζ(x0)∈ζ[(A∪{x0})∩(F∪{x0})]⊆ζ[U].
By Lemma 3.3, there exists E∈ℱr(W) such that
(3.14)(B∪{x0})∩(E∪{x0})⊆ζ[(A∪{x0})∩(F∪{x0})].
Let  G=(B∪{x0})∩(E∪{x0}).  Note that  E∪{x0}∈τ.  Hence,  G∈ρ  and
(3.15)b=x0∈G⊆ζ[(A∪{x0})∩(F∪{x0})]⊆ζ[U].Case 2.
Assume thatb∈B.  Consequently, {b}∈𝒫(X∖{x0}), which implies {b}∈τ, which implies  {b}∈ρ (since {b}={b}∩[B∪{x0}]). Therefore, b∈ζ[U]  implies
(3.16)b∈{b}⊆ζ[U].
From expression (3.15) in Case 1 and expression (3.16) in Case 2,  we infer
(3.17)ζ[U]∈ρ.
LetS∈ρ.  We will show that ζ-1[S]∈λ.
Leta∈ζ-1[S].  Either a=x0 or a∈A.Case 3.
Assume thata=x0.  Hence,  x0=ζ(x0)=ζ(a)∈S.  So, there exists M∈ℬ1  such that x0∈(B∪{x0})∩M⊆S.  Since  x0∉D  for any  D∈𝒫(X∖{x0}),  we have that M=H∪{x0}  for some  H∈ℱr(W).  Therefore,
(3.18)(B∪{x0})∩M=(B∪{x0})∩(H∪{x0}),
which implies
(3.19)x0∈(B∪{x0})∩(H∪{x0})⊆S,
which implies
(3.20)a=ζ-1(x0)∈ζ-1[(B∪{x0})∩(H∪{x0})]⊆ζ-1[S].
By Lemma 3.3, there exists C∈ℱr(W) such that
(3.21)(A∪{x0})∩(C∪{x0})⊆ζ-1[(B∪{x0})∩(H∪{x0})].
Let  T=(A∪{x0})∩(C∪{x0}).  Note that  C∪{x0}∈τ.  Hence,  T∈λ  and
(3.22)a=x0∈T⊆ζ-1[(B∪{x0})∩(H∪{x0})]⊆ζ-1[S].Case 4.
Assume thata∈A.  Consequently,  {a}∈𝒫(X∖{x0}), which implies {a}∈τ, which implies  {a}∈λ (since {a}={a}∩[A∪{x0}]).  Therefore,   a∈ζ-1[S]  implies
(3.23)a∈{a}⊆ζ-1[S].
From expression (3.22) in Case 3 and expression (3.23) in Case 4,  we infer that
(3.24)ζ-1[S]∈λ.
Consequently,  from expression (3.17) and (3.24), we infer that ζ is a homeomorphism of  A∪{x0}  onto B∪{x0}.Proposition 3.5.
Letτ be the Hausdorff topology on X generated by ℬ1.  Let  K⊆X be a compact set.  If  W⊆K, then  W∪{x0} is homeomorphic to K.Proof.
Note thatW⊆K implies |W|≤|K|, which implies K in an infinite set (W is an infinite set), which implies  K=A∪{x0} for some A⊆X∖{x0} such that  A∩V  is a finite set by Proposition 2.11.  Consequently,  W⊆K  and x0∉W  imply  W⊆A.  So,
(3.25)A=A∩(W∪V)=W∪(A∩V),
which implies
(3.26)|A|=|W|+|A∩V|=|W|,
(see [2, Corollary  2.3, page 162]). Therefore, W∪{x0} is homeomorphic to A∪{x0}=K by Proposition 3.4.Remark 3.6.
LetZ be a nonempty set and let  θ be a Hausdorff topology on Z.  If  A⊆Z  is a nonempty finite set, then θ induces the discrete topology on A.Theorem 3.7.
Letτ be the Hausdorff topology on X generated by ℬ1, and let K1 and K2 be nonempty compact subsets of  X.  If  K1 and K2  have the same cardinality (i.e., |K1|=|K2|), then there exists a homeomorphism of K1 onto K2.Proof.
LetK1 and K2 be nonempty compact subsets of  X such that |K1|=|K2|. If  K1 is a finite set, then K2 is a finite set.  Hence, τ induces the discrete topology on K1 and K2 (see Remark 3.6). Consequently, any bijection of  K1 onto K2 is a homeomorphism.
Assume thatK1 is an infinite set. Hence, K2 is an infinite set. So, K1=A∪{x0} and  K2=B∪{x0} such that  A⊆X∖{x0},  B⊆X∖{x0},  A∩V  is a finite set and  B∩V  is a finite set  by Proposition 2.11. Observe that K1 and K2 being infinite sets imply A and B are infinite subsets of  X∖{x0} such that |A|=|B|. Therefore, there exists a homeomorphism of K1 onto K2 by Proposition 3.4.


## 4. Examples

Example 4.1.
LetW={1/n}n=1∞,  V=∅, and x0=0. Let
(4.1)X=V∪W∪{x0}={1n}n=1∞∪{0},
and let
(4.2)B1=P({1n}n=1∞)∪{F∪{x0}∣F∈Fr({1n}n=1∞)}.
Consider the Hausdorff space  (X,τ), where the topology τ  is generated by ℬ1. Observe that  (X,τ) is compact (Corollary 2.10, since V=∅) and W={1/n}n=1∞ is not compact by Corollary 2.5.  If  K⊆X  is an infinite compact set, then
(4.3)|K|=ω=|{1n}n=1∞∪{0}|,
therefore, K  is homeomorphic to {1/n}n=1∞∪{0} (Theorem 3.7). In other words, all infinite, compact subsets of {1/n}n=1∞∪{0} are homeomorphic. Note that topology  τ  on  X  is induced by the standard euclidean topology on ℝ.Example 4.2.
LetB be a set such that  |B|=ω. Let  φ:ω→B be a bijection. For n∈ω, we will denote φ(n)  by  xn,  that is,  xn=φ(n). Therefore,  B={xn}n∈ω, where m,n∈ω  and  m≠n  imply  xm≠xn.  Let  W={x2j}j∈ℕ  and let  V={x2j+1}j∈ω. Note that the maps  j→x2j  and   j→x2j+1  are bijections of  ℕ onto W and ω onto V, respectively. Also,  x0∉W∪V,  W∩V=∅  and  B={x0}∪V∪W; consequently, V=[B∖{x0}]∖W. We can write ℬ1⊆𝒫(B) (see Definition 2.2) as follows. 							(4.4)B1=P({xn}n=1∞)∪{F∪{x0}∣F∈Fr(W)}.
The collection  ℬ1  generates a Hausdorff topology τ on B (see Proposition 2.3). Note that  (B,τ) is not compact (Corollary 2.10), sets V and W are not compact (Corollaries 2.6 and 2.5, resp.), V∪{x0} is not compact (Corollary 2.9), and W∪{x0} is compact (Corollary 2.8). Also, if K⊆B is an infinite compact set, then
(4.5)|K|=ω=|W∪{x0}|,
therefore,  K  is homeomorphic to W∪{x0}={x2j}j∈ω (Theorem 3.7). In other words, all infinite, compact subsets of B={xn}n∈ω are homeomorphic.Example 4.3.
Letθ be an infinite cardinal and consider the collection {κn}n∈ω of infinite cardinals defined as follows.  Let  κ0=ω  and for n∈ω, let  κn+1=2κn.  Also, we denote  κω=⋃n∈ωκn.  Hence,  κω is the cardinal number, that is, the supremum of  {κn}n∈ω  and  κn<κω for each n∈ω (see the Alephs section in [3], page 29).  For n∈ω, define
(4.6)Wn={(μ,n)∣μ∈κn},Wω={(μ,ω)∣μ∈κω}.
Observe that |Wω|=κω,  |Wn|=κn  for each n∈ω, and the collection
(4.7){Wn}n∈ω∪{Wω}
is pairwise disjoint.  Let
(4.8)W=Wω∪[⋃n∈ωWn],V={(λ,ω+1)∣λ∈θ},x0=(2κω,ω+2).
Note that the collection {{x0},W,V}  is pairwise disjoint. Let
(4.9)X={x0}∪V∪W
and consider the noncompact, Hausdorff space (X,τ), where topology τ is generated by  ℬ1 (see Definition 2.2 and Corollary 2.10).  For η∈ω+1, define
(4.10)Kη={K∈P(X)∣Kiscompact,|K|=κη}.
Observe,  Wη∪{x0}∈𝒦η  for  η∈ω+1  by Proposition 2.7 (since  Wη∩V=∅) and the fact that |Wη∪{x0}|=κη.  Also, the collection {𝒦η}η∈ω+1 is pairwise disjoint and Theorem 3.7  implies that all of the sets in 𝒦η are homeomorphic to each other. Since  κn<κω for each n∈ω, we have
(4.11)|W|=|Wω∪[⋃n∈ωWn]|=κω+κω=κω,
(see [2, Corollary  2.3, page 162]), which implies  |W∪{x0}|=κω.  Consequently, W∪{x0}∈𝒦ω (see Corollary 2.8), which implies W∪{x0} is homeomorphic to Wω∪{x0}. If we let [V]<ω denote the set of all finite subsets of  V,  then
(4.12)(Wη∪E)∪{x0}∈KηforE∈[V]<ω,η∈ω+1,
(see Proposition 2.7). Consequently, the size of θ can affect the cardinality of 𝒦η  for  η∈ω+1 (e.g., let θ be a Mahlo cardinal ([3, Chapter 8, page 95])). If the generalized continuum hypothesis is assumed, then the collection {𝒦η}η∈ω+1 is a partition of the collection of all infinite, compact subsets of X.



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