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@ Appl. Gen. Topol. 17, no. 2(2016), 117-122doi:10.4995/agt.2016.4521
c© AGT, UPV, 2016

Results about the Alexandroff duplicate space

Khulod Almontashery and Lutfi Kalantan

King Abdulaziz University, Department of Mathematics, P.O.Box 80203, Jeddah 21589, Saudi

Arabia (khuloodalim@hotmail.com, LK274387@hotmail.com and lkalantan@kau.edu.sa)

Abstract

In this paper, we present some new results about the Alexandroff Du-

plicate Space. We prove that if a space X has the property P , then its
Alexandroff Duplicate space A(X) may not have P , where P is one of
the following properties: extremally disconnected, weakly extremally
disconnected, quasi-normal, pseudocompact. We prove that if X is α-
normal, epinormal, or has property wD, then so is A(X). We prove
almost normality is preserved by A(X) under special conditions.

2010 MSC: 54F65; 54D15; 54G20.

Keywords: Alexandroff duplicate; normal; almost normal; mildly normal;

quasi-normal; pseudocompact; rroperty wD; α-normal; epinor-

mal.

There are various methods of generating a new topological space from a given
one. In 1929, Alexandroff introduced his method by constructing the Double
Circumference Space [1]. In 1968, R. Engelking generalized this construction
to an arbitrary space as follows [6]: Let X be any topological space. Let
X′ = X × {1}. Note that X ∩ X′ = ∅. Let A(X) = X ∪ X′. For simplicity,
for an element x ∈ X, we will denote the element 〈x, 1〉 in X′ by x′ and for
a subset B ⊆ X, let B′ = {x′ : x ∈ B} = B × {1} ⊆ X′. For each x′ ∈ X′,
let B(x′) = {{x′}}. For each x ∈ X, let B(x) = {U ∪ (U′ \ {x′}) : U is open
in X with x ∈ U }. Then B = {B(x) : x ∈ X} ∪ {B(x′) : x′ ∈ X′} will
generate a unique topology on A(X) such that B is its neighborhood system.
A(X) with this topology is called the Alexandroff Duplicate of X. Now, if P
is a topological property and X has P , then A(X) may or may not have P .
Throughout this paper, we denote an ordered pair by 〈x, y〉, the set of positive
integers by N and the set of real numbers by R. For a subset A of a space X,

Received 09 January 2016 – Accepted 03 August 2016

http://dx.doi.org/10.4995/agt.2016.4521


K. Almontashery and L. Kalantan

intA and A denote the interior and the closure of A, respectively. An ordinal
γ is the set of all ordinal α such that α < γ. The first infinite ordinal is ω and
the first uncountable ordinal is ω1.

A topological space X is called α-normal [3] if for any two disjoint closed
subsets A and B of X, there exist two open disjoint subsets U and V of X such
that A ∩ U dense in A and B ∩ V dense in B.

Theorem 0.1. If X is α-normal, then so is its Alexandroff Duplicate A(X).

Proof. Let E and F be any two disjoint closed sets in A(X). Write E = E1∪E2,
where E1 = E ∩ X, E2 = E ∩ X

′ and F = F1 ∪ F2, where F1 = F ∩ X,
F2 = F ∩ X

′. So, we have E1 and F1 are two disjoint closed sets in X. By
α-normality of X, there exist two disjoint open sets U and V of X such that
E1 ∩U is dense in E1 and F1 ∩V is dense in F1. Let W1 = (U ∪U

′ ∪E2)\F and
W2 = (V ∪V

′ ∪F2)\E. Then W1 and W2 are disjoint open sets in A(X). Now,
we prove W1 ∩ E is dense in E. Note that W1 ∩ E = (W1 ∩ E1) ∪ (W1 ∩ E2) =

(U ∩E1)∪E2, hence W1 ∩ E = (U ∩ E1) ∪ E2 = (U ∩ E1)∪E2 ⊃ E1 ∪E2 ⊃ E.
Therefore, W1 ∩ E is dense in E. Similarly, W2 ∩ F is dense in F . Therefore,
A(X) is α-normal. �

A space X is called extremally disconnected [5] if it is T1 and the closure
of any open set is open. Extremally disconnectedness is not preserved by the
Alexandroff Duplicate space and here is a counterexample.

Example 0.2. Consider the Stone-Čech compactification space βω which is
compact Hausdorff, hence Tychonoff. It is well-known that βω is extremally
disconnected. Clearly ω is open in A(βω) and ωA(βω) = βω which is not open
in A(βω).

A space X is called weakly extremally disconnected [8] if the closure of any
open set is open. Weakly extremally disconnected is not preserved and the
above example is a counterexample. The following question is interesting and
still open: “ Does there exist a Tychonoff non-discrete space X such that A(X)
is extremally disconnected?”.

A subset B of a space X is called a closed domain [5] if B = intB. A finite
intersection of closed domains is called π-closed [9]. A topological space X
is called mildly normal [7] if for any two disjoint closed domains A and B of
X, there exist two open disjoint subsets U and V of X such that A ⊆ U and
B ⊆ V . A topological space X is called quasi-normal [9] if for any two disjoint
π-closed subsets A and B of X, there exist two open disjoint subsets U and
V of X such that A ⊆ U and B ⊆ V . It is clear from the definitions that
every quasi-normal space is mildly normal. Mild normality is not preserved by
the Alexandroff Duplicate space [7]. Quasi-normality is not preserved by the
Alexandroff Duplicate space and here is a counterexample. We denote the set
of all limit points of a set B by Bd and call it the derived set of B.

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Results about the Alexandroff duplicate space

Example 0.3. Consider Rω1, which is Tychonoff separable non-normal space
([5], 2.3.15). It is well-known that every closed domain in Rω1 depends on
a countable set [12]. It follows that every π-closed set in Rω1 depends on a
countable set. Now, if A and B are disjoint π-closed sets in Rω1, then there is
a countable S ⊂ ω1 such that A = πS(A) × R

ω1\S and B = πS(B) × R
ω1\S,

where πS is the projection function πS : R
ω1 −→ RS. It follows that πS(A) and

πS(B) are disjoint closed sets in R
S. Since S is countable, RS is metrizable.

So, there exist two open disjoint sets U1, V1 ⊂ R
S such that πS(A) ⊆ U1 and

πS(B) ⊆ V1. Then A ⊆ U = U1 × R
ω1\S and B ⊆ V = V1 × R

ω1\S where U
and V are open in Rω1 and disjoint. Therefore, Rω1 is quasi-normal.

We show that the Alexandroff Duplicate space A(Rω1) is not quasi-normal
by showing that it is not mildly normal. Let

E = {〈nξ : ξ < ω1〉 ∈ N
ω1 : ∀ m ∈ N \ {1}(|{ξ < ω1 : nξ = m}| ≤ 1) }.

F = {〈nξ : ξ < ω1〉 ∈ N
ω1 : ∀ m ∈ N \ {2}(|{ξ < ω1 : nξ = m}| ≤ 1) }.

E and F are disjoint closed subsets in Nω1, hence closed in Rω1. They cannot
be separated by disjoint open sets, see [14], and they are perfect, i.e., E = Ed

and F = F d. By a theorem from [7] which says: “If A and B are disjoint
subsets of a space X such that Ad and Bd cannot be separated, then A(X) is
not mildly normal.”, we conclude that A(Rω1) is not mildly normal.

A space X is pseudocompact [5] if X is Tychonoff and any continuous real-
valued function defined on X is bounded. Equivalently, a Tychonoff space X is
pseudocompact if and only if any locally finite family consisting of non-empty
open subsets is finite [5]. We will conclude that pseudocompactness is not
preserved by the Alexandroff Duplicate space by the following theorem.

Theorem 0.4. Let X be a Tychonoff space. The Alexandroff Duplicate A(X)
is pseudocompact if and only if X is countably compact.

Proof. If X is not countably compact, then there exists a countably infinite
closed discrete subset D of X. Then D × {1} = D′ is closed and open set in
A(X) which is also countably infinite, hence A(X) is not pseudocompact. Now,
assume that X is countably compact. Since for a set B ⊆ X and a point a ∈ X,
we have that a is a limit point for B′ if and only if a is a limit point for B, we
do have that A(X) is also countably compact and hence pseudocompact. �

So, any pseudocompact space which is not countably compact will be an
example of a pseudocompact space whose Alexandroff Duplicate space A(X)
is not pseudocompact. A Mrówka space Ψ(A) , where A ⊂ [ω]ω is maximal, is
such a space, see [4] and [10].

Arhangel’skii introduced the notions of epinormality and C-normality in
2012, when he was visiting the department of mathematics, King Abdulaziz
University, Jeddah, Saudi Arabia. A topological space X is called C-normal
[2] if there exist a normal space Y and a bijective function f : X −→ Y such

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K. Almontashery and L. Kalantan

that the restriction f|C : C −→ f(C) is a homeomorphism for each compact
subspace C ⊆ X. It was proved in [2] that if X is C-normal, then so is its
Alexandroff Duplicate. A topological space ( X , τ ) is called epinormal [2] if
there is a coarser topology τ ′ on X such that ( X , τ ′ ) is T4.

Theorem 0.5. If X is epinormal, then so is its Alexandroff Duplicate A(X).

Proof. Let X be any space which is epinormal, let τ be a topology on X, since
X is epinormal, then there is a coarser topology τ∗ on X such that ( X , τ∗ ) is
T4. Since T4 is preserved by the Alexandroff Duplicate space, then A(X , τ

∗ )
is also T4 and it is coarser than A(X , τ) by the topology of the Alexandroff
Duplicate. Hence, A(X) is epinormal. �

A space X is said to satisfy Property wD [11] if for every infinite closed
discrete subspace C of X, there exists a discrete family {Un : n ∈ ω} of open
subsets of X such that each Un meets C at exactly one point.

Theorem 0.6. If X satisfies property wD, then so does its Alexandroff Dupli-

cate A(X).

Proof. Let X be any space which satisfies property wD and consider its Alexan-
droff Duplicate A(X). To show that A(X) has Property wD, let C ⊆ A(X) be
any infinite closed discrete subspace of A(X). Write C = (C ∩ X) ∪ (C ∩ X′).
For each x ∈ C ∩ X, fix an open set Ux in X such that Vx = Ux ∪ (U

′
x \ {x

′})
open in A(X) and

(*) Vx ∩ C = {x}

Case 1: C ∩ X is finite. This implies that C ∩ X′ is infinite. Let {x′n : n ∈
ω} ⊆ C ∩ X′ such that x′i 6= x

′
j, for all i, j ∈ ω with i 6= j. Now, consider the

family {{x′n}: n ∈ ω}, then it consists of open sets and each {x
′
n} meets C at

exactly one point. Now, we will show that {{x′n}: n ∈ ω} is a discrete family
in A(X). It is obvious that {x′n : n ∈ ω} is discrete and it is closed because if
x ∈ A(X) \ C, then there is open set Ux containing x such that Ux ∩ C = ∅,
and if x ∈ C ∩ X hence, by (*) there is an open set Vx in A(X) containing x
such that Vx ∩ C = {x}. Thus, {{x

′
n}: n ∈ ω} is discrete family. Therefore, in

this case, A(X) satisfies property wD.
Case 2: C ∩ X is infinite. Then C ∩ X is an infinite closed discrete subspace

of X. Since X satisfies the property wD, then there exists a discrete family
{Vn : n ∈ ω} of open subsets of X such that each Vn meets C ∩ X at exactly
one point {xn}. Hence, {Vn ∪(V

′
n \ {x

′
n}): n ∈ ω} is discrete in A(X) and then

meets C in exactly one point {xn}. Therefore, also in this case, A(X) satisfies
the property wD. �

A space X is called almost normal [8] if for any two disjoint closed subsets
A and B of X one of which is a closed domain there exist two disjoint open
sets U and V of X such that A ⊆ U and B ⊆ V .

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Results about the Alexandroff duplicate space

Proposition 0.7. Let X be a topological space. The following are equivalent:

(1) X is the only non-empty closed domain in X.
(2) Each non-empty open subset is dense in X.
(3) The interior of each non-empty proper closed subset of X is empty.

Proof. (1) =⇒ (2) Let U be any non-empty open set. Suppose U is not dense
in X, then U is a non-empty proper closed domain in X, which contradicts the
hypothesis, hence U is dense in X.
(2) =⇒ (3) Suppose that E is a non-empty proper closed subset of X such that
intE 6= ∅, then X = intE ⊆ E = E which is a contradiction.
(3) =⇒ (1) Suppose that there exists a closed domain B such that ∅ 6= B 6= X,
then ∅ 6= intB 6= X , which contradicts the hypothesis, thus X is the only non-
empty closed domain in X. �

It is clear that any space that satisfies the conditions of Proposition 7 will
be almost normal.

Corollary 0.8. If X satisfies the conditions of Proposition 7, then its Alexan-

droff Duplicate A(X) is almost normal.

Proof. Let E and F be any non-empty disjoint closed subsets of A(X) such
that E is a closed domain. Let W be an open set in A(X) such that W = E.
If W ∩ X 6= ∅, then W ∩ X is dense in X by Proposition 7, so X ⊂ W = E.
It follows that F ⊂ X′, so E is closed and open, hence there are two disjoint
open sets U = A(X) \ F and V = F in A(X) containing E and F respectively.

If W ∩ X = ∅, then E ⊂ X′, so E is closed and open, thus there are two
disjoint open sets U = E and V = A(X) \ E in A(X) containing E and F
respectively. �

Corollary 0.9. The Alexandroff Duplicate of the countable complement space

[13], the finite complement space [13], and the particular point space [13] are
all almost normal.

The following problems are still open:

(1) If X is almost normal, is then its Alexandroff Duplicate A(X) almost
normal?

(2) If X is β-normal [3], is then its Alexandroff Duplicate A(X) β-normal?

Acknowledgements. The authors would like to thank the referee for valuable

suggestions and comments.

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K. Almontashery and L. Kalantan

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