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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 14, no. 1, 2013

pp. 17-32

On star compactifications

Lorenzo Acosta and I. Marcela Rubio

Abstract

We study the ordered structure of the collection of star compactifica-

tions by n points and the behavior of these compactifications through

quotients obtained by identification of additional points.

2010 MSC: Primary 54D35, 54A10. Secondary 54D60, 54B15

Keywords: Star compactifications, quotient spaces, ordered structure.

1. Introduction.

When we study compactification in a general topology course, we usually only
deal with three types of compactifications: (1) the Alexandroff compactifica-

tion, which is a one-point compactification; (2) the Stone C̆ech’s compactifica-
tion, and (3) some elementary examples of compactifications by a finite number
of points, for instance:

(1) [0, 1] is a compactification of (0, 1) by two points.
(2) [0, 1] ∪ [2, 3] is a compactification of A = (0, 1) ∪ (2, 3) by four points.
(3) Geometrically, two disjoint circles on R2 are a compactification by two

points, of the set A in the previous example.

These examples of compactifications by a finite number of points are parti-
cular cases of star compactifications. Star topologies were defined in [7] and
we mention without proofs some results given in [7]; in particular, the ne-
cessary and sufficient conditions for such topologies to be compactifications or
T2−compactifications of a non-compact space.



18 L. Acosta and I. M. Rubio

By compactness of a topological space X, we mean that every open covering of
X has a finite subcovering, and by compactification of a non-compact topo-
logical space X, not necessarily a Hausdorff space, we mean a compact space
containing X as a subspace, which is dense in the compactification.

Our intention is to illustrate the definition with simple star compactifications
and present some interesting results about the ordered structure of the collec-
tion of star compactifications by n points, of a topological space. We obtain the
stability of such collection by finite intersections. On the other hand, we present
the behavior of the quotients of star compactifications obtained through equi-
valence relations in which we make some identification between the additional
points. Moreover we establish the relationships between the star compactifica-
tions of a topological space and other compactifications of it by a finite number
of points.

2. Preliminary results.

2.1. Star topologies and star compactifications. In this section we present
the definition and some known results concerning star topologies and star com-
pactifications, given in [7].

For this purpose, we consider a non-compact topological space (X, τ) and Xn =
X ∪ {ω1, ..., ωn} , where ω1, ..., ωn are n distinct points not belonging to X,
n ∈ N.

Proposition 2.1. Let Ui, i = 1, ..., n be open subsets contained in X.
Then

B = τ ∪ {(Ui \ K) ∪ {ωi} | K ⊆ X closed and compact; i = 1, ..., n}

is a base for a topology µ on Xn.

This topology is called the star topology associated to U1, ..., Un.

Notice that B is a closed collection under finite intersections. Throughout
this paper we denote the star topology over Xn associated to U1, ..., Un by
µ = 〈〈U1, ..., Un〉〉.

Proposition 2.2. (Xn, µ) is a compactification of (X, τ) if and only if

(1) X \
n
∪
i=1

Ui is compact, and

(2) Ui * K for each K closed and compact subset of X, i = 1, ..., n.

Observe that the second condition implies that for each i = 1, ..., n, Ui is non-
empty. By the definition of µ we observe that (X, τ) is a subspace of (Xn, µ)
and X ∈ µ.



On star compactifications 19

2.2. The Alexandroff compactification. Let (X, τ) be a non-compact topo-
logical space and X1 = X ∪ {ω} , where ω is a point not belonging to X.

Theorem 2.3. If η = τ∪{A ∪ {ω} | A ∈ τ and X \ A is compact} , then (X1, η)
is a compactification of (X, τ) by one point.

This compactification is called the Alexandroff compactification of (X, τ) .
It is the finest compactification of (X, τ) obtained by adding one point, and it
is called “the” compactification by one point in the class of Hausdorff spaces,
because it is the only Hausdorff compactification by one point when (X, τ) is
a Hausdorff and locally compact space. See [6].

3. Ordered structure of the star compactifications by n points.

We present some results regarding the inclusion order relation in the collection
of star compactifications by n points of a non-compact topological space, which
enable us to conclude that this collection is stable under finite intersections.
We exhibit the smallest element of the mentioned collection. This element can
be seen as a generalization of the Alexandroff compactification by n points,
with n > 1. On the other hand, we study the relationship between the open
sets that generate two star topologies when one of them is finer than the other.

Let (X, τ) be a non-compact topological space and W = {ω1, ω2, ω3, ...} be a
set of different elements not belonging to X.
We denote X0 = X, Xn = Xn−1 ∪ {ωn} , n ≥ 1.

It is known that all compactifications of (X, τ) by n points are seen, up to
homeomorphisms, as Xn with a convenient topology, in such way that X is
considered a subspace of the compactification.

We denote

En = {µ ∈ T op(Xn) | (Xn, µ) is a star compactification of (X, τ)} ;

the collection of star compactifications of (X, τ) by n points. Observe that for
each µ ∈ En, we have τ ⊂ µ, i.e., X ∈ µ.
Notice that the inclusion order relation defined in En coincides with the order
usually defined between compactifications (see [4]).

Proposition 3.1. Let Ω = 〈〈U1, ..., Un〉〉 , where Ui = X for each i. If µ =
〈〈V1, ..., Vn〉〉 is an arbitrary star topology on Xn, then Ω ⊆ µ.

Proof. Let K be a closed and compact subset of (X, τ). As Vi ⊆ X for each
i = 1, ..., n, then A = {ωi} ∪ (X \ K) = {ωi} ∪ (Vi \ K) ∪ (X \ K) ∈ µ, since
{ωi} ∪ (Vi \ K) ∈ µ and X \ K ∈ τ ⊆ µ. Therefore Ω ⊆ µ. �

This proposition asserts that Ω is the smallest element of the set of star topolo-
gies on Xn, ordered by inclusion. Since Ω satisfies the mentioned conditions in
Proposition 2.2, then Ω is a star compactification of (X, τ) by n points and Ω
is the smallest element of (En, ⊆).



20 L. Acosta and I. M. Rubio

Remark 3.2. For the case n = 1, Ω = 〈〈X〉〉 = B, where

B = τ ∪ {{ω1} ∪ (X \ K) | K ⊆ X closed and compact} ,

we have that Ω = B and Ω is the Alexandroff compactification of (X, τ). Since
the Alexandroff compactification is the finest compactification of (X, τ) by one
point, then Ω is the only star compactification of (X, τ) by one point.

Proposition 3.3. If µ = 〈〈V1, ..., Vn〉〉 is a star compactification of (X, τ)
where V1 = V2 = ... = Vn, then µ = Ω.

Proof. Let {ωi} ∪ (Vi \ K) be a basic open set in µ for some i = 1, ..., n and
some K closed and compact subset of X. X \ Vi is closed and compact since
n
∪
j=1

Vj = Vi.

Thus

{ωi} ∪ (Vi \ K) = {ωi} ∪ [X \ (K ∪ (X \ Vi))] ∈ Ω,

and µ ⊆ Ω. �

Remark 3.4.

(1) If µ = 〈〈V1, ..., Vn〉〉 is an arbitrary star topology on Xn with V1 = V2 =
... = Vn, it could happen that µ * Ω.
Example: Consider X = (0, 1) with the usual topology as subspace of
R; V1 = ... = Vn =

(

1
4
, 1
2

)

. Then {ωi} ∈ µ, but {ωi} /∈ Ω.
(2) Proposition 3.3 is a particular case of Theorem 0.8 in [7]: “Let A1, ..., An

and B1, ..., Bn be two n-tuples of open sets in X. The star topologies
associated with those two n-tuples are the same if and only if the sets
Ai − Bi and Bi − Ai are contained in compact and closed sets for all
i = 1, ..., n”.

Proposition 3.5. Let µ = 〈〈U1, ..., Un〉〉 , β = 〈〈V1, ..., Vn〉〉 be two star topolo-
gies on Xn. If Vi ⊆ Ui for each i = 1, ..., n, then µ ⊆ β.

Proof. Let A = {ωi} ∪ (Ui \ K) ∈ µ. Since Vi ⊆ Ui and Ui \ K ∈ τ, then
A = {ωi} ∪ (Vi \ K) ∪ (Ui \ K) ∈ β. �

The next proposition asserts that the insersection of two star topologies on Xn
is of the same kind and it describes the open sets associated with it.

Proposition 3.6. Let µ = 〈〈U1, ..., Un〉〉 , β = 〈〈V1, ..., Vn〉〉 be two star topolo-
gies on Xn, then µ ∩ β = η where η = 〈〈U1 ∪ V1, ..., Un ∪ Vn〉〉 .

Proof. i) By the previous proposition we have that η ⊆ µ because Ui ⊆ Ui∪Vi,
for each i = 1, ..., n and for the same reason, η ⊆ β.
ii) For the other inclusion it is enough to see that all basic open neighborhoods
of ωi in µ ∩ β are open neighborhoods of ωi in η, for each i = 1, ..., n.



On star compactifications 21

Let M be a basic open neighborhood of ωi in µ ∩ β, for i = 1, ..., n, that is,
M = {ωi} ∪ (Ui \ K1) ∪ A = {ωi} ∪ (Vi \ K2) ∪ B, where K1 and K2 are closed
and compact subsets of X and A, B ∈ τ.
Since (Ui ∪ Vi) \ (K1 ∪ K2) ⊆ (Ui \ K1) ∪ (Vi \ K2) and
(Ui \ K1) ∪ A = (Vi \ K2) ∪ B, then

(Ui \ K1) ∪ A = (Ui \ K1) ∪ (Vi \ K2) ∪ A ∪ B

= [(Ui ∪ Vi) \ (K1 ∪ K2)] ∪ (Ui \ K1) ∪ (Vi \ K2) ∪ A ∪ B.

So, if we call C = (Ui \ K1) ∪ (Vi \ K2) ∪ A ∪ B, then C is an open set of τ and
therefore M = {ωi} ∪ [(Ui ∪ Vi) \ (K1 ∪ K2)] ∪ C ∈ η. �

Proposition 3.7. Let µ = 〈〈U1, ..., Un〉〉 , β = 〈〈V1, ..., Vn〉〉 be two star topolo-
gies on Xn. If µ ⊆ β then there exist Mi, Ni open sets of τ such that Ni ⊆ Mi
for each i = 1, ..., n and µ = 〈〈M1, ..., Mn〉〉 , β = 〈〈N1, ..., Nn〉〉 .

Proof. Let Ni = Vi, Mi = Ui ∪ Vi for each i = 1, ..., n sets of τ. Therefore
µ = 〈〈M1, ..., Mn〉〉 , β = 〈〈N1, ..., Nn〉〉 and Ni ⊆ Mi for each i = 1, ..., n. �

Observe that this proposition is a weak version of the reciprocal of Proposition
3.5.

The next proposition guarantees that intersection is a closed operation in the
collection En of star compactifications of (X, τ) by n points.

Proposition 3.8. If µ = 〈〈U1, ..., Un〉〉 , β = 〈〈V1, ..., Vn〉〉 are two star com-
pactifications of (X, τ) by n points, then η = 〈〈U1 ∪ V1, ..., Un ∪ Vn〉〉 is a star
compactification of (X, τ) by n points.

Proof. i) We have that Ui ∪ Vi * K for each closed and compact subset K of
X, because Ui * K for each i = 1, ..., n.
ii)

X \
n
∪
i=1

(Ui ∪ Vi) = X \
((

n
∪
i=1

Ui

)

∪
(

n
∪
i=1

Vi

))

=
(

X \
n
∪
i=1

Ui

)

∩
(

X \
n
∪
i=1

Vi

)

,

where X \
n
∪
i=1

Ui and X \
n
∪
i=1

Vi are closed and compact subsets of X, then

X \
n
∪
i=1

(Ui ∪ Vi) is compact. Therefore, η is a star compactification of (X, τ)

by n points. �

4. Certain quotients of star compactifications.

We consider the quotients obtained by an equivalence relation ⋄ on Xn; ⋄ =
{(x, x) | x ∈ Xn} ∪ R , where R is an equivalence relation on {ω1, ..., ωn}. The
star compactifications of these quotients have an interesting behavior.

Definition 4.1. We say that a compactification (Y, µ) of (X, τ) is of A-class
if X ∈ µ.



22 L. Acosta and I. M. Rubio

Observe that star compactifications are of A-class.

The next theorem asserts that a quotient of A-class compactification of (X, τ),
obtained through the mentioned equivalence relation ⋄, is an A-class compac-
tification of (X, τ).

Theorem 4.2. Let (Xn, µ) be an A-class compactification of (X, τ) by n points,
n > 1. If we consider the equivalence relation ⋄ on Xn defined above, then
(Xn/⋄, µ/⋄) is an A-class compactification of (X, τ) by m points, where m =
|{ω1, ..., ωn} /R| ≤ n.

Proof. We have X
i
→֒ Xn

θ
−→ Xn/⋄, where i is the topological imbedding and

θ is the standard quotient map.
i) (Xn/⋄, µ/⋄) is compact because θ : (Xn, µ) −→ (Xn/⋄, µ/⋄) is a surjective
continuous map, where (Xn, µ) is a compact space.

ii) To see that θ◦i(X) is dense in Xn/⋄ we need to see that θ ◦ i(X)
µ/⋄

= Xn/⋄
= {[x] | x ∈ X} ∪ {[ωi] | i = 1, ..., n} .
If A = {[ωi] | i = 1, ..., n} , then |A| = |{ω1, ..., ωn} /R| = m.

Supose that θ ◦ i(X)
µ/⋄

$ Xn/⋄, that is, that there exists B ∈ P (A) \ {∅}
such that B ∈ µ/⋄, then θ−1(B) ∈ P ({ω1, ..., ωn})\{∅} and θ−1(B) ∈ µ, but

this is a contradiction because X is dense in Xn; therefore θ ◦ i(X)
µ/⋄

= Xn/⋄
and θ ◦ i(X) is dense in Xn/ ⋄ .

iii) Let us see that θ ◦ i : X → Xn/⋄ is a homeomorphism between X and
θ ◦ i(X), where θ ◦ i(X) = θ(X) = {[x] = {x} | x ∈ X} :
i is a homeomorphism between X and i(X) = X. By the definition of ⋄, θ
is a bijective map between i(X) = X and θ(X), then θ ◦ i is a bijective map
between X and θ ◦ i(X) = θ(X). By the continuity of θ we have that θ ◦ i is
continuous.
Let us see that θ ◦ i is an open map:
Let A be an open set of τ, we need to see that θ ◦ i(A) = θ(A) ∈ µ/ ⋄ .
θ(A) = {[a] | a ∈ A} = {{a} | a ∈ A} because A ⊂ X.
θ(A) ∈ µ/⋄ if and only if θ−1 (θ(A)) ∈ µ. Since θ(A) ⊂ θ(X) and θ is a bijective
map between X and θ(X), then θ−1 (θ(A)) = A. Since i is a homeomorphism
between X and i(X) = X and A ∈ τ, then i(A) = A = B ∩ X, where B ∈ µ
and since X ∈ µ then A ∈ µ and θ(A) ∈ µ/ ⋄ . Thus, θ ◦ i is an open
map, and then, it is a homeomorphism between X and θ ◦ i(X). Moreover,
θ(X) ≈ X ∈ µ/⋄ because X ∈ µ.
Hence (Xn/⋄, µ/⋄) is an A-class compactification of (X, τ) by m points, where
m = |{ω1, ..., ωn} /R| ≤ n. �

Corollary 4.3. (Xn/⋄, µ/⋄) is homeomorphic to an A-class compactification
of (X, τ) by m points, seen as (Xm, η) for an appropriate η.



On star compactifications 23

Consider (Xn, µ) a star compactification of (X, τ) by n points, with
µ = 〈〈U1, ..., Un〉〉 . Since (Xn, µ) is of A-class, we know that (Xn/⋄, µ/⋄) is a
compactification of (X, τ) by m points, where m = |{ω1, ..., ωn} /R| ≤ n.
Since the quotient map θ : Xn −→ Xn/⋄ is bijective in X, we say θ (x) = x
and θ (A) = A for each x ∈ X and for each A ⊆ X, and then Xn/⋄ = X ∪ V
with V = {υ1, ..., υm} = {θ (ωi) | i = 1, ..., n} , υ1 = θ (ω1) , υk = θ (ωi) where
i = min {j ∈ {1, ..., n} | ωj /∈ υ1 ∪ ... ∪ υk−1} for each k = 2, ..., m.
We denote Ii = {j ∈ {1, ..., n} | θ (ωj) = υi} for each i = 1, ..., m and if |Ii| =
ni, then Ii = {i1, ..., ini} .

The next proposition asserts that (Xn/⋄, µ/⋄) is a star compactification of
(X, τ).

Proposition 4.4. µ/⋄ = 〈〈M1, ..., Mm〉〉 where Mi =
ni
∪

j=1
Uij for each i =

1, ..., m.

Proof. Let A be a basic open set of 〈〈M1, ..., Mm〉〉: If A ∈ τ, since θ
−1(A) = A

and τ ⊆ µ then A ∈ µ/ ⋄ . If A = {υi} ∪ (Mi \ K) for some i = 1, ..., m and
some K closed and compact subset of X, then
θ−1(A) = {ωi1, ..., ωini} ∪ (Mi \ K)

= {ωi1, ..., ωini} ∪

((

ni
∪

j=1
Uij

)

\ K

)

= {ωi1, ..., ωini} ∪

(

ni
∪

j=1
(Uij \ K)

)

=
ni
∪

j=1

({

ωij
}

∪ (Uij \ K)
)

∈ µ,

then A ∈ µ/ ⋄ . Thus 〈〈M1, ..., Mm〉〉 ⊆ µ/ ⋄ .

Consider now A ∈ µ/⋄, we see that A ∈ 〈〈M1, ..., Mm〉〉 by showing that all its
points are interior in 〈〈M1, ..., Mm〉〉 .
We know that θ−1(A) ∈ µ and let z be an element of A :
Case 1: If z ∈ X then z ∈ θ−1(A) and since µ |X= τ, there exists an open set
θ−1(A) ∩ X = B ∈ τ such that z ∈ B ⊆ θ−1(A) then z ∈ θ(B) = B ⊆ A. Since
B ∈ 〈〈M1, ..., Mm〉〉 then z is an interior point of A in 〈〈M1, ..., Mm〉〉 .
Case 2: z = υi for some i = 1, ..., m.
θ−1(υi) = {ωi1, ..., ωini} ⊆ θ

−1(A) and θ−1(A) is an open set of µ, then for
each j ∈ Ii, there exists

{

ωij
}

∪ (Uij \ Kj) open set of µ and subset of θ
−1(A).

Since

(

ni
∪

j=1
Uij

)

\

(

ni
∪

j=1
Kj

)

⊆
ni
∪
j=1

(Uij \ Kj) we have:

E =
ni
∪

j=1

[{

ωij
}

∪ (Uij \ Kj)
]

= θ−1(υi) ∪

[(

ni
∪

j=1
Uij

)

\

(

ni
∪

j=1
Kj

)]

∪
ni
∪

j=1
(Uij \ Kj)

= θ−1(υi) ∪ (Mi \ K) ∪ B ⊆ θ
−1(A),

where K =
ni
∪

j=1
Kj is a closed and compact subset of X and

B =
ni
∪

j=1
(Uij \ Kj) is an open set of τ.



24 L. Acosta and I. M. Rubio

Thus, υi ∈ θ (E) = {υi} ∪ (Mi \ K) ∪ B ⊆ A and since
{υi} ∪ (Mi \ K) ∪ B is open in 〈〈M1, ..., Mm〉〉, then υi is an interior point of
A in 〈〈M1, ..., Mm〉〉 .
By cases 1 and 2 we conclude that A ∈ 〈〈M1, ..., Mm〉〉 . �

Therefore, the quotient of a star compactification by n points is a star com-
pactification by m points, where n ≥ m. Consequently and by Remark 3.2, if
⋄ = R ∪ {(x, x) | x ∈ X} , where R is the equivalence relation on {ω1, ..., ωn}
in which all these points are related, then (Xn/⋄, µ/⋄) is the Alexandroff com-
pactification of (X, τ) .

Moreover, by observing the form of the open sets Mi and in view of the Propo-
sition 3.6 we obtain the next corollary.

Corollary 4.5. µ/⋄ = ∩
p∈P

〈〈Up1, ..., Upm〉〉 where P = I1 × ... × Im and

p = (p1, ..., pm) .

For instance, if ⋄ only identifies ω1 with ω2 on Xn then, in Xn−1 we have:

µ/ ⋄ = 〈〈U1 ∪ U2, U3, ..., Un〉〉 = 〈〈U1, U3,..., Un〉〉 ∩ 〈〈U2, U3,..., Un〉〉 .

On the other hand, the star compactifications show an interesting behavior
when, from a star compactification of (X, τ) by m points, we obtain a com-
pactification of (X, τ) by n points, with n ≥ m, so that certain quotient on Xn
gives back the original compactification.

In terms of Theorem 4.2 we have the next proposition.

Proposition 4.6. If (Xm, µ) with µ = 〈〈U1, ..., Um〉〉 is a star compactification
of (X, τ) by m points, then (Xn, β) with

β = µ ∪ {B ∪ A | B ⊆ {ωm+1, ..., ωn} , A ∈ µ , {ω1} ∪ A ∈ µ}

is a star compactification of (X, τ) by n points, n ≥ m, such that a certain
quotient of (Xn, β) is (Xm, µ) .

Proof. Let see that β = 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 .
By the definition of 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 we observe that
µ ⊆ 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 .
Let C be a basic open of 〈〈U1, U2, ..., Um, U1, ..., U1〉〉:
If C ∈ τ or C = {ωi} ∪ (Ui \ K) for some i = 1, ..., m and some K closed and
compact subset of X, then C ∈ µ ⊆ β.
If C = {ωi} ∪ (U1 \ K) for some i = m + 1, ..., n and some closed and compact
subset K of X, we have that A = U1 \ K ∈ τ ⊆ µ and {ω1} ∪ A ∈ µ, thus
C = B ∪ A with B = {ωi} ⊆ {ωm+1, ..., ωn} , then C ∈ β.
Therefore 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 ⊆ β.
On the other hand let C be an element of β. We see that
C ∈ 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 showing that all of its points are interior points
in 〈〈U1, U2, ..., Um, U1, ..., U1〉〉.



On star compactifications 25

Let z be a point of C :
Case 1: If z ∈ Xm, by the definition of β we have that C ∩ Xm ∈ µ, thus
z ∈ C ∩ Xm ⊆ C where C ∩ Xm ∈ 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 and z is an
interior point of C in 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 .

Case 2: If z = ωi, for some i = m + 1, ..., n, then C = B ∪ A where ωi ∈ B,
A ∈ µ, {ω1}∪A ∈ µ, thus there exists a closed and compact subset K of X such
that {ω1}∪(U1 \K) ⊆ {ω1}∪A. Since U1 \K ⊆ A, then ωi ∈ {ωi}∪(U1 \K) ⊆
B ∪ A = C where {ωi} ∪ (U1 \ K) is open in 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 .
Therefore ωi is an interior point to C in 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 .
By cases 1 and 2 we conclude that C ∈ 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 .
Finally, in the presentation of β as 〈〈U1, U2, ..., Um, U1, ..., U1〉〉 it is clear that
the equivalence relation on Xn that identifies ω1, ωm+1, ..., ωn produces the star
topology µ. �

Remark 4.7. From a star compactification of µ = 〈〈U1, ..., Um〉〉 on Xm it is
quite simple to generate new compactifications by n points, n > m, by repeating
the Ui as the open sets associated to new points.

For instance, β = 〈〈U1, U2, ..., U7, U4, U3〉〉 is a star compactification of (X, τ)
by nine points obtained from the star compactification by seven points µ =
〈〈U1, ..., U7〉〉 of (X, τ). Moreover, if ⋄ is the equivalence relation on X9 that
identifies ω4 with ω8 and ω3 with ω9 then (X9/⋄, β/⋄) is the compactification
(X7, µ) .

5. Star compactifications vs Magill compactifications.

A Magill compactification refers to the method of compactification by n points
of a non-compact topological space, presented by Magill in [3]. In this section
we show the relation between Magill compactifications and star compactifica-
tions.

5.1. The Magill compactifications. Using the notation that we have intro-
duced in this paper, we present the Magill compactification with his results
without proofs.

Proposition 5.1. If X contains n non-empty open subsets Gi, i = 1, ..., n;
two by two disjoint such that:

(1) H = X \
n
∪
i=1

Gi is compact and

(2) X \ ∪
j 6=i

Gj is not compact for each i = 1, ..., n,

then the collection

B∗ = τ ∪ {A ∪ {ωi} | A ∈ τ, (H ∪ Gi) ∩ (X \ A) is compact in X; i = 1, ..., n}

is a base for a topology ρ on Xn.

Observe that H ∪ Gi = X \ ∪
j 6=i

Gj for each i = 1, ..., n because Gi are mutually

disjoint. Under these conditions we have the following propositions.



26 L. Acosta and I. M. Rubio

Proposition 5.2. (Xn, ρ) is a compactification of (X, τ) by n points.

Proposition 5.3. If (X, τ) is locally compact and T2 then (Xn, ρ) is T2.

Remark 5.4. Although Magill always considers Hausdorff topological spaces
in [3], it is clear that this construction still provides a compactification of X
without this assumption.

5.2. Relation between star compactifications and Magill compactifi-
cations. In this section we obtain that each Magill compactification of X by
n points is a star compactification. Thus the collection of Magill compactifica-
tions of X by n points coincides with the collection of star compactifications
with associated two by two disjoint open sets. In the case of Hausdorff compac-
tifications, when X is Hausdorff and locally compact, we have that the collec-
tions of Magill compactifications of X by n points and of star compactifications
of X by n points coincide and are all the possible Hausdorff compactifications
of X by n points.

Proposition 5.5. Each star compactification of (X, τ) by n points, with asso-
ciated two by two disjoint open sets is a Magill compactification of (X, τ) by n
points.

Proof. Let µ be a star compactification 〈〈U1, ..., Un〉〉 of (X, τ) where the sets
Ui are two by two disjoint. The sets Ui are open, non-empty subsets of X and

H = X \
n
∪
i=1

Ui is compact.

Moreover, X \ ∪
j 6=i

Uj is not compact because in the contrary case, since

Ui ⊆ X \ ∪
j 6=i

Uj, with X \ ∪
j 6=i

Uj a subset of X closed and compact, this contra-

dicts that (Xn, µ) is a compactification of (X, τ) .
Therefore, the sets Ui, i = 1, ..., n produce a Magill compactification that we
call ρ.
We assert that µ = ρ.
i) Let (Ui \ K) ∪ {ωi} be an element of B, base of the star compactification µ,
where K is a closed and compact subset of X.
To see that (Ui \ K) ∪ {ωi} ∈ B

∗, base of the Magill compactification ρ, we
show that (H ∪ Ui) ∩ (X \ (Ui \ K)) is compact:
(H ∪ Ui) ∩ (X \ (Ui \ K))

=

(

X \ ∪
j 6=i

Uj

)

∩ (X \ (Ui ∩ (X \ K)))

= X \

((

∪
j 6=i

Uj

)

∪ (Ui ∩ (X \ K))

)

= X \

((

n
∪
j=1

Uj

)

∩

((

∪
j 6=i

Uj

)

∪ (X \ K)

))

=

(

X \
n
∪
j=1

Uj

)

∪

(

X \

((

∪
j 6=i

Uj

)

∪ (X \ K)

))

=

(

X \
n
∪
j=1

Uj

)

∪

((

X \ ∪
j 6=i

Uj

)

∩ K

)

,



On star compactifications 27

where X \
n
∪

j=1
Uj is compact and so is

(

X \ ∪
j 6=i

Uj

)

∩ K because K is compact

and X \ ∪
j 6=i

Uj is closed. Thus (H ∪ Ui) ∩ (X \ (Ui \ K)) is compact, with

Ui \ K ∈ τ; then B ⊆ B
∗ and µ ⊆ ρ.

ii) Let A ∪ {ωi} be an element of B
∗, base of ρ, then (H ∪ Ui) ∩ (X \ A) is

compact and closed. Thus Ui \ [(H ∪ Ui) ∩ (X \ A)] ∪ {ωi} ∈ B.
Ui \ [(H ∪ Ui) ∩ (X \ A)] = [Ui \ (H ∪ Ui)] ∪ [Ui \ (X \ A)]

=

[

Ui \

(

X \ ∪
j 6=i

Uj

)]

∪ [Ui \ (X \ A)]

= ∅ ∪ [Ui \ (X \ A)]
= A ∩ Ui.

Since A ∈ τ, we have that A∪{ωi} = A∪[(A ∩ Ui) ∪ {ωi}] ∈ µ and ρ ⊆ µ. �

Proposition 5.6. Each Magill compactification of (X, τ) by n points is a star
compactification of (X, τ) by n points.

Proof. Let ρ be a Magill compactification of (X, τ) obtained through n open,
non-empty subsets of X, Gi, i = 1, ..., n two by two disjoint.
Let µ be the star topology 〈〈G1, ..., Gn〉〉. Reasoning as in the previous propo-
sition we obtain that ρ = µ. Thus ρ is a star compactification of X and we
have that Gi * K, for each K closed and compact subset of X. �

So we have that the collection of Magill compactifications of X by n points:
Mn and the collection of star compactifications of X by n points associated
to n two by two disjoint open sets: EDn are the same. From this fact and
Proposition 4.4 we obtain the following corollary.

Corollary 5.7. If (Xn, µ) is a Magill compactification of (X, τ) by n points and
⋄ = R ∪ {(x, x) | x ∈ X}, where R is an equivalence relation on {ω1, ..., ωn},
then (Xn/⋄, µ/⋄) is a Magill compactification of (X, τ) by m points, where
m = |{ω1, ..., ωn} /R| ≤ n.

Remark 5.8. There exist star compactifications that are not Magill compacti-
fications.
Let X be the subspace (0, 1) of R with the usual topology, and X2 = X ∪
{ω1, ω2} , µ = 〈〈U1, U2〉〉 where U1 =

(

0, 1
2

)

, U2 = (0, 1) .
It is clear that (X2, µ) is a star compactification of X by two points. If µ is a
Magill compactification, then by Proposition 5.6 we have that µ = 〈〈G1, G2〉〉
where G1 and G2 are non-empty, open and disjoint subsets of X, that satisfy
the other conditions to be a compactification. But since 〈〈G1, G2〉〉 is a com-
pactification of X by two points, with G1 and G2 disjoints, it follows that there
are a, b ∈ (0, 1) such that G1 = (0, a) , G2 = (b, 1) with a < b; without loss of
generality we can suppose that a < 1

2
< b then, by Proposition 3.5 we have

that 〈〈U1, U2〉〉 ⊆ 〈〈G1, G2〉〉 .



28 L. Acosta and I. M. Rubio

Moreover, G2 ∪ {ω2} ∈ 〈〈G1, G2〉〉 but G2 ∪ {ω2} /∈ 〈〈U1, U2〉〉 because for all
A ∪ {ω2} ∈ 〈〈U1, U2〉〉 there exists c ∈ (0, 1) such that (0, c) ⊆ A and clearly
(0, c) * G2.
Therefore 〈〈U1, U2〉〉 $ 〈〈G1, G2〉〉 and (X2, µ) is a star compactification of X
that is not a Magill compactification.
Furthermore, for each n > 1 we have that Mn $ En because (Xn, Ω)1 is a star
compactification of (X, τ) that is not a Magill compactification.

We mention without proof Theorem 0.5 of [7]:
“Let X be a locally compact Hausdorff space. Then any Hausdorff compactifi-
cation for X is the star topology associated with an m− tuple of open mutually
disjoint subsets of X”.

We denote:
HCn the collection of T2 compactifications of X by n points,
HEn the collection of T2 star compactifications of X by n points and
HMn the collection of T2 Magill compactifications of X by n points.

Proposition 5.9. If (X, τ) is T2, locally compact and non-compact then
HCn = HEn = HMn.

Proof. i) Clearly HEn ⊆ HCn. By Theorem 0.5 of [7] we have HCn ⊆ HEn.
ii) By Proposition 5.6, HMn ⊆ HEn. Let µ be an element of HEn ⊆ HCn, by
Theorem 0.5 of [7], µ is a T2 star compactification of X associated with n open
mutually disjoint subsets of X. Then µ ∈ HMn by Proposition 5.5. �

Remark 5.10. Observe that µ = 〈〈U1, U2〉〉 is not a T2 compactification of X,
by Remark 5.8.

6. Some examples.

In this section we present simple examples of star compactifications, and exam-
ples of compactifications of a topological space by a finite number of points that
are not of A-class and, therefore, are not star compactifications.

Example 6.1. Consider the topological space (R, τ) , where

τ = {(−x, x) : x > 0} ∪ {φ, R} .

To find the star compactifications by two points of this space we need two open
sets U1, U2 such that L = R\ (U1 ∪ U2) is compact. Since the only closed and
compact set of this space is φ we need that U1 ∪ U2 = R and this happens
if one of them is R. Thus, for this space we have basically three types of star
compactifications by two points: 〈〈U, R〉〉 , 〈〈R, U〉〉 and 〈〈R, R〉〉 , where U is
an open set of the form (−x, x) .

On the other hand, there exist infinity star compactifications by two points of
type 〈〈U, R〉〉 , that depend on the open set U = (−x, x) that we consider. All

1Considered as defined in Proposition 3.1.



On star compactifications 29

the compactifications of this type are ordered by inclusion, so that each non-
empty subcollection of them has its intersection or its union in the same type
of star compactifications. That is, we again obtain a star compactification.
However, in general this fact is false.

Example 6.2. Let X be the open interval (0, 1) of R with the usual topology
of subspace. In this case there exist seven star compactifications by two points,
which are classified in three types: (1) the lowest compactification 〈〈X, X〉〉 ;
(2) four of the type 〈〈X, U2〉〉 or 〈〈U1, X〉〉 that are obtained accordingly if
Ui, for i = 1, 2 is either (0, a) or (a, 1) for some a, with 0 < a < 1; and (3)
two maximal compactifications of the form 〈〈U1, U2〉〉 , where U1 = (0, a) and
U2 = (b, 1) or on the contrary, where a, b ∈ (0, 1) and in this case the order
between a and b is irrelevant because the different possibilities produce the
same compactification.

The two maximal compactifications can be seen as the compactification [0, 1]
of (0, 1) where the additional points 0 and 1 correspond to ω1 and ω2; ω1 = 0
if U1 = (0, a) , or ω1 = 1 if U1 = (b, 1) .

Example 6.3. Consider X = (0, 1) ∪ (2, 3) as a subspace of R with the usual
topology. In this case the star compactifications by two points are classified
in twelve different types. In these twelve, we find three types of maximal
compactifications in which X2 can be represented as a subset of R2, where
the basic neighborhoods of ω1 and ω2 are precisely obtained by the subspace
topology of R2, with the usual topology. These can be represented as in Figure
1:

(X2, µ) (X2, ρ) (X2, ϕ)

ω1

ω2

•

•

....

....

....

....

....

....

.....
.....
.....
.....
......
......
.......
........

..........
...................................................................................................................................................................................................................................................
..........
........
.......
......
......
.....
.....
.....
.....
.....
....
....
....
....
. ω1

ω2• • .....
....
....
.....
.....
.....
......
.......
........

........................................................................................................................................................................

........
.......
......
.....
.....
.....
....
....
....
......................................................................................

ω1 ω2
• •

....

....

....

.....
.....
.....
.....
......
.......
...........

....................................................................................................................................................................
........
......
......
.....
.....
.....
....
....
....
.....

....

....

....

.....
.....
.....
......
.......
.........

........................................................................................................................................................................
........
......
......
.....
.....
.....
....
....
....
..

X2 is the X2 is the union of X2 is the union of
circumference. the segment and the two disjoint

the circumference. circumferences.

Figure 1: Maximal compactifications.

Some intermediate compactifications are:

• η = 〈〈U1, U2〉〉 , where U1 = (0, 1) and U2 = (b, 1)∪(2, 3) , with 0 < b <
1. We have that η ⊂ ϕ and η ⊂ ρ.

• γ = 〈〈V1, V2〉〉 , where V1 = (0, 1) ∪ (2, a) and V2 = (b, 1) ∪ (2, 3) , with
2 < a < 3 and 0 < b < 1. We have that γ ⊂ η.



30 L. Acosta and I. M. Rubio

• α = 〈〈W1, W2〉〉 , where W1 = (0, a) ∪ (2, 3) and W2 = (b, 1) ∪ (2, 3) ,
with 0 < a < 1 and 0 < b < 1. Notice that α and γ are incomparable.

We know that every star compactification is of A-class, but there exist com-
pactifications of A-class that are not star compactifications. This is illustrated
in the following example.

Example 6.4. Consider the topological space (X, τ) of the real numbers with
the usual topology and the simplest possible compactification: (Xn, µ) , where
µ = τ ∪{Xn} , for n ≥ 1. We can easily verify that (Xn, µ) is a compactification
of A-class of (X, τ) that is not star.

The following propositions provide examples of non A-class compactifications.

Proposition 6.5. If (X1, µ) is a compactification of (X, τ) by one point, where
X1 = X ∪ {ω} , then τ ⊆ µ.

Proof. i) X ∈ µ.
In fact, since (X, τ) is not compact, there exists a covering {Ai | i ∈ I} of X
by open sets of τ that cannot be reduced to a finite covering.
For each i ∈ I there exists Bi ∈ µ such that Ai = Bi ∩ X.

X1 = ∪
i∈I

Bi =⇒ X1 =
n
∪

k=1
Bik (because X1 is compact).

=⇒ X =
n
∪

k=1
(Bik ∩ X) =

n
∪

k=1
Aik.

This is a contradiction and then X1 6= ∪
i∈I

Bi.

But

X1 % ∪
i∈I

Bi =⇒ ω /∈ Bi, for all i ∈ I

=⇒ Bi ⊆ X, for all i ∈ I
=⇒ Ai = Bi, for all i ∈ I
=⇒ X ∈ µ.

ii) A ∈ τ =⇒ A = B ∩ X for some B ∈ µ
=⇒ A ∈ µ. (because X ∈ µ.) �

The previous proposition asserts that X is an open set in every compactification
of (X, τ) by one point. This is equivalent to saying that each element of τ is an
element of µ. Thus, each compactification of (X, τ) by one point is obtained
with the original open sets and by adding some additional sets in an appropriate
way. However, this fact cannot be generalized to compactifications by more
than one point.

Definition 6.6. We say that a topological space (Y, υ) is hyperconnected if
υ \ {∅} is a collection closed for finite intersections, that is to say, if each pair
of non-empty open sets has non-empty intersection.



On star compactifications 31

Some examples:

(1) (Y, ϕ) , where Y is an infinite set and ϕ is the cofinite topology, is
hyperconnected.

(2) (R, τ), where τ is the topology with base {(−a, a) ⊂ R |a > 0}, is hy-
perconnected.

(3) (R, µ), where µ is the topology with base {(a, +∞) ⊂ R |a ∈ R}, is
hyperconnected.

(4) R with the usual topology is not a hyperconnected space.

The next proposition provides examples of compactifications by more than one
point that are not of A-class.

Proposition 6.7. If (X, τ) is a hyperconnected, non-compact topological space
then (X, τ) has a compactification by more than one point in which X is not
an open set.

Proof. Let X2 be the set X ∪ {ω1, ω2}

µ = {A ∪ {ω1} | A ∈ τ \ {∅}} ∪ {∅, X2} .

It is clear that µ is a topology on X2 and X /∈ µ.
i) (X2, µ) is compact because if {Bi | i ∈ I} is a covering of X2 by open sets
of µ, X2 = Bi, for some i ∈ I, and then this covering can be reduced to a finite
one.
ii) µ |X= τ.
iii) X is dense in X2, because on the contrary some {ω1} , {ω2} or {ω1, ω2}
will be open sets of µ.
Thus, (X2, µ) is a compactification of (X, τ) by more than one point, in which
X is not a open set. �

Remark 6.8. In the previous proposition we can take Y = X ∪Z, where ω1 ∈ Z,
X ∩Z = ∅ and Z is a finite or infinite set that contains more than one element.
Following the proof we obtain that (Y, µ) is a compactification of (X, τ) by more
than one point that does not contain X as an open set.

The next fact can be proved in a simple way.

Proposition 6.9. If (Xn, µ) is a T1 compactification of (X, τ) then (Xn, µ) is
of A-class.

Question 6.10. If we consider other types of compactifications of (X, τ) by n
points, what is the relationship between these types of compactifications and
Star (Magill) compactifications?

Acknowledgements. We would like to thank professors Lucimar Nova and
Januario Varela for their useful comments.



32 L. Acosta and I. M. Rubio

References

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nal of Undergraduate Mathematics, Furman University. 2 (1996), 1–5.

[2] N. Bourbaki, Elements of mathematics, General topology, Addison-Wesley, France, 1968.
[3] K. D. Magill Jr., N-point compactifications, Amer. Math. Monthly. 72 (1965), 1075-

1081.
[4] J. Margalef et al. Topoloǵıa, vol. 3. Alhambra, Madrid, 1980.
[5] J. R. Munkres, Topology. A first course, Prentice-Hall, New Jersey, 1975.
[6] M. Murdeshwar, General topology, John Wiley and Sons, New York, 1983.
[7] T. Nakassis and S. Papastavridis, On compactifying a topological space, by adding a

finite number of points, Bull. Soc. Math., Grece. 17 (1976), 59-65.
[8] C. Rúız and L. Blanco, Acerca del compactificado de Alexandroff, Bol. Mat. (3) 20

(1986), 163-171.
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(Received August 2011 – Accepted November 2012)

L. Acosta (lmacostag@unal.edu.co)
Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de
Matemáticas, Carrera 30 No. 45-03, Bogotá, Colombia.

I. M. Rubio (imrubiop@unal.edu.co)
Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de
Matemáticas, Carrera 30 No. 45-03, Bogotá, Colombia.


	On star compactifications. By L. Acosta and I. M. Rubio