Topology and its Applications 161 (2014) 196–205

Contents lists available at ScienceDirect

Topology and its Applications

www.elsevier.com/locate/topol

I-spaces, nodec spaces and compactifications

Karim Belaid a,∗, Lobna Dridi b

a University of Dammam, Faculty of Sciences of Dammam, Girls College, Department of Mathematics,PO Box 383, Dammam 31113, Saudi Arabiab University of Monastir, Higher Institute of Mathematics and Informatics (ISIM), Department ofMathematics, PO Box 223, 5000 Monastir, Tunisia

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 December 2012Received in revised form 7 July 2013Accepted 10 October 2013

MSC:primary 06E15, 54F65secondary 54D35

Keywords:Nodec spaceI-spaceOne point compactificationWallman compactificationHerrlich compactification

In this paper, we describe compact nodec spaces and we characterize space such thatits one point compactification (respectively Wallman compactification) is nodec. Wealso establish a characterization of spaces such that their compactification is anI-space. And we give necessary and sufficient conditions on the space X in order toget its Herrlich compactification remainder finite.

© 2013 Elsevier B.V. All rights reserved.

Introduction

A subset N of a topological space X is called nowhere dense if the interior of the closure of N is theempty set. Recall that a space X is a nodec space if each nowhere dense subset of X is closed. A topologicalspace X is an I-space if its derived set Xd (that is the set of accumulation points) is discrete. And, if everysubset of X is an intersection of a closed subset and an open set of X, then the X is said to be submaximal.Submaximal spaces have been studied by several authors (see, for instance, [1,7,9]).

Classically we have the following implications:

I-space ⇒ Submaximal ⇒ Nodec

It is shown in [3] that a compactification K(X) of a topological space X is submaximal if and only if foreach dense subset D of X, the following properties hold:

* Corresponding author.E-mail addresses: belaid412@yahoo.fr (K. Belaid), lobna.dridi@gnet.tn (L. Dridi).

0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.topol.2013.10.021

K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205 197

(i) K(X)\D is finite.(ii) For each x ∈ K(X) \D, {x} is closed.

The first section of this paper contains some remarks of nodec spaces and compact nodec spaces. We alsogive a necessary and sufficient conditions on a space X in order to get its one point compactification (resp.Wallman compactification) nodec.

The second section deals with the characterization of spaces such that their compactification are I-spaces.We establish a necessary and sufficient conditions on a space X to get its one point compactification (resp.Wallman compactification) an I-space.

In the third section we give a characterization of spaces such that their Herrlich compactification remain-der is finite.

Throughout this paper we consider spaces on which no separation axioms are assumed unless explicitlystated. Let X be a topological space and A be a subset of X. The closure of A in X is denoted by clX(A),and if A is finite we denote Card(A) the cardinality of A.

1. Nodec spaces and compactifications

Let first recall the definition of the Krull dimension of a T0-space. Let (X, T ) be a T0-space. Then X hasa partial order �, induced by T by taking x � y if and only if y ∈ clX(x). Hence clX(x) = {y | y � x} isthe specialization of x [8]. The notion of the Krull dimension defined on the prime spectrum of a ring hasbeen generalized to T0-spaces [4]. The chain x0 < x1 < · · · < xn of elements of X is said to be a chain oflength n. The supremum of the lengths of chains is called the Krull dimension of (X, T ) and is denoted bydimK(X, T ).

In [2] the authors have proved that if a T0-space (X, T ) is submaximal, then dimK(X, T ) � 1.

Proposition 1.1. Let (X, T ) be a nodec T0-space. Then dimK(X, T ) � 1.

Proof. Let x ∈ X. If y ∈ clX(x)\{x}, then {y} is a nowhere dense subset of X. Hence {y} is closed. Thusy is a maximal point (for the order induced by the topology T ) of X. Therefore dimK(X, T ) � 1. �Example 1.2. There exists a nodec space with Krull dimension 1 which is not a submaximal space.

Let Z be the set of the integers, equipped with the topology T = {∅}∪{U ⊆ Z | 0 ∈ U and Z\U is finite}.Clearly dimK(Z, T ) = 1, whilst (Z, T ) is nodec, since every nowhere dense set is finite. On the other hand,(Z, T ) is not submaximal, since {0} is not an intersection of a closed subset and an open subset of Z.

In fact, more can be said.

Example 1.3. There exists a T0-space with Krull dimension 1 which is not a nodec space.Let X = Z ∪ {ω1, ω2} and � be the order on X defined by 2n + 1 � 2n, 2n− 1 � 2n and ω1 � ω2. Let

U be a collection of upper subsets X (that is, A ∈ U if and only if {y ∈ X | x � y} ⊆ A, for each x ∈ A).Let T be the topology on X whose closed sets are empty set and elements A of A such that A is finite orω2 ∈ A. Let L = {2n | n ∈ Z}. Then clX(L) = L ∪ {ω2} and the interior of clX(L) is the empty set. Hence,L is a nowhere dense subset of X but not a closed set of X. Thus X is not nodec.

We need the following definition to describe compact nodec spaces.

Definition 1.4. A topological space X is said to be a strong-nodec space (s-nodec, for short) if each nowheredense subset is finite and closed.

198 K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205

Proposition 1.5. Let X be a nodec space. If C is a compact nowhere dense set of X, then C is finite.

Proof. Let x ∈ C. Since C − {x} is a nowhere dense set of X, C − {x} is a closed set of X. Hence {x} isan open set of C. Thus C is finite, since C is compact. �Proposition 1.6. Let X be a compact space. Then the following statements are equivalent:

(1) X is nodec;(2) X is s-nodec.

Proof. (1) =⇒ (2) Let N be a nowhere dense subset N of X. Then N\{n} is also a nowhere dense subsetof X, for each n ∈ N . Hence (X\N) ∪ {n} is an open set of X. Thus {n} is an open subset of N . That N

is finite follows immediately from the fact that N is a compact closed set of X. Then X is s-nodec.(2) =⇒ (1) Straightforward. �Recall that a compactification of a topological space X is a compact space K(X) such that X is homeo-

morphic to a dense subspace of K(X).The following lemma is a consequence of Proposition 1.6.

Lemma 1.7. Let X be a topological space and K(X) a compactification of X. If K(X) is nodec then X iss-nodec.

Proof. Let N be a nowhere dense set of X. The interior of clX(N) is empty. Let O be an open set of K(X)such that O ⊆ clK(X)(N). Hence O∩X ⊆ clK(X)(N)∩X. Since clX(N) = clK(X)(N)∩X, O∩X is empty;so O is empty. Thus N is a nowhere dense subset of K(X). Then N is finite and {n} is closed set of K(X),for each n ∈ N . Indeed, {n} is a closed set of X. Therefore X is s-nodec. �

The must famous compactification is the Alexandroff compactification: Let X be a topological space, setX̃ = X ∪ {∞} with the topology whose members are the open subsets of X and all subsets U of X̃ suchthat X̃\U is a closed compact set of X. The space X̃ is called the Alexandroff extension of X. If X is anon-compact space, then X̃ is a compactification called the Alexandroff compactification (or the one pointcompactification of X).

Proposition 1.8. Let X be a non-compact topological space. Then the Alexandroff compactification of X isnodec if and only if X is s-nodec.

Proof. Necessary condition. Follows immediately from Lemma 1.7.Sufficient condition. Let N be a nowhere dense subset of the Alexandroff compactification X̃ of X. We

consider two cases.Case 1: N = {∞}. In this case, N is closed.Case 2: N ∩X = ∅. In this case, N ∩X is a nowhere dense subset of X. Since X is s-nodec, N ∩X is a

finite closed set of X; so N ∩X is a compact closed set of X. Hence N ∩X is a closed set of X̃. Since N isequal to N ∩X or (N ∩X) ∪ {∞}, N is a closed set of X̃. �

We give an example of nodec space such that its Alexandroff compactification is not nodec.

Example 1.9. Let L be an infinite set and a /∈ L. Set X := L ∪ {a}, and equip X with the topologyT = {∅} ∪ {O ⊆ X | a ∈ O}. Of course, (X, T ) is an Alexandroff T0-space with Krull dimension 1. Thus,(X, T ) is nodec. In fact, it is straightforward to see that the nowhere dense sets of X are exactly the closed

K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205 199

sets of X. However, the Alexandroff compactification X̃ of X is not nodec (since L is an infinite nowheredense set of X).

We turn our attention to spaces such that their Wallman compactification are nodec spaces.First, let us recall the Wallman compactification construction: Let X be a T1-space and P be the class

of closed sets of X. A closed filter is a collection F of nonempty elements of P with the properties:

(i) F is closed under finite intersections;(ii) P1 ∈ F and P1 ⊆ P2 implies P2 ⊆ F .

A closed ultrafilter is a maximal closed filter. Let wX be the collection of all closed ultrafilters on X.If U is an open set of X, set U∗ = {F ∈ wX | F ⊆ U for some F in F}. If C is a closed set of X, setC∗ = {F ∈ wX | C ∈ F}.

In [10], Wallman has proved the following result:

Proposition 1.10. Let X be a T1-space, C and G be two closed sets of X. Then the following statementshold:

(1) wX − C∗ = (X − C)∗.(2) (C ∪G)∗ = C∗ ∪G∗.(3) (C ∩G)∗ = C∗ ∩G∗.

Hence the collection C = {D∗ | D is a closed set of X} is a base for closed sets of a topology T on wX,and it is easily seen that the class {U∗ | U is an open set of X} is a base for open sets of T .

Let ϕ : X → wX be the map defined by ϕ(x) = {C | C is a closed set of X and x ∈ C}. Then ϕ is anembedding. Since ϕ(X) is a dense set of wX, X will be identified with ϕ(X). Hence X is assumed to be adense subset of wX.

Finally, since wX is a compact space, wX is a compactification of the T1-space X called the Wallmancompactification of X. The set wX −X is called the Wallman compactification remainder. Thus elementsof wX −X are closed ultrafilter F such that

⋂(F : F ∈ F) = ∅.

In 1995, Kovar [6] has characterized space such that its Wallman compactification remainder is finite(that is, Card(wX\X) < +∞), he proved the following theorem.

Theorem 1.11. Let X be a T1-space. The following statements are equivalent:

(i) There exists k ∈ N such that any pairwise disjoint family of closed sets in X contains at most k

non-compact elements;(ii) The Wallman compactification remainder of X is finite.

We need further new concepts.

Definition 1.12. Let X be a topological space, H be a collection of nonempty closed sets of X satisfying thefinite intersection property and n ∈ N.

– Call H an n-collection if each collection F of non-compact closed sets such that F ∩ H = ∅, for eachF ∈ F and H ∈ H, has at most n elements.

– Call H a special collection (sp-collection, for short) if Card(⋂

(H: H ∈ H)) is finite and {x} is not open,for each x ∈

⋂(H: H ∈ H).

200 K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205

Now, we are in a position to give a characterization of spaces such that their Wallman compactificationis nodec.

Proposition 1.13. Let X be a T1-space. Then the following statements are equivalent:

(1) The Wallman compactification of X is nodec;(2) X satisfies the following properties:

(i) X is s-nodec.(ii) For each sp-collection H, there exists n ∈ N such that H is an n-collection.

Proof. (1) =⇒ (2) That X is s-nodec follows immediately from Lemma 1.7.Let H be an sp-collection of X and K =

⋂(H∗: H ∈ H). Since

⋂(H: H ∈ H) = K ∩X and X is T1,

K is a nowhere dense subset of wX. So K is finite, since wX is s-nodec (Proposition 1.6).Let k = Card(K) and suppose that there exists a collection C of k + 1 pairwise disjoint non-compact

closed sets of X such that F ∩H = ∅, for each H ∈ H and F ∈ C.Let F ∈ C. Since for every H ∈ H, F ∩H = ∅, there exists a closed ultrafilter F such that F ∈ F and

H ⊆ F . Hence F ∈ K, since F ∈ H∗, for each H ∈ H. On the other hand, C is a collection of k+1 pairwisedisjoint non-compact closed sets of X; so {F | F ∈ C} is a collection of pairwise distinct element of K. ThenCard(K) � k + 1. This is impossible, since k = Card(K). Therefore there exists n � k such that H is ann-collection.

(2) =⇒ (1) Let N be a nowhere dense subset of wX.Step 1: clwX(N) ∩X is a nowhere set of X.Suppose that there exists an open set O of X such that O ⊆ clwX(N) ∩ X. Let F ∈ wX such that

F ∈ O∗. Let U be an open set of X such that F ∈ U∗. Then U∗ ∩O∗ = ∅. So that U ∩ clwX(N) = ∅, sinceU ∩ O = ∅. Hence F ∈ clwX(N). Thus O∗ ⊆ clwX(N), contradicting the fact that N is nowhere dense setof wX. Therefore clwX(N) ∩X is a nowhere dense set of X.

Step 2: N closed.Since clwX(N) is closed, there exists a collection H of a nonempty closed sets of X such that clwX(N) =⋂

(H∗: H ∈ H). Hence Card(⋂

(H: H ∈ H)) = Card(clwX(N) ∩X) < +∞, since X is s-nodec, and {x} isnot open, for each x ∈

⋂(H: H ∈ H).

Thus H is an sp-collection and, by hypothesis (ii), there exists n ∈ N such that H is an n-collection.Set A = (wX\X)∩ clwX(N) and suppose that Card(A) � n+1. Then there exists a collection K of n+1

pairwise disjoint non-compact closed sets of X such that, for each F ∈ K, there exists F ∈ A and F ∈ F .Since A ⊆

⋂(H∗: H ∈ H), one obtains immediately that F ∈ H∗, for each H ∈ H and F ∈ F ; so

F ∩H = ∅, and that contradicts the fact that H is an n-collection. Hence Card(A) < n+ 1. Thus clwX(N)is finite. Therefore N is closed. �2. I-spaces and compactifications

Our goal in the present section is to characterize space such that its compactification is an I-space.First, we need the following definition to describe I-spaces.

Definition 2.1. Let X be a topological space. A subset A of X is said to be a closed-finite space (c-finite,for short) if A is finite and {a} is closed, for each a ∈ A.

Let us state a useful lemma.

Lemma 2.2. Let X be a compact topological space. Then the following statements are equivalent.

K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205 201

(1) X is an I-space;(2) Xd is a c-finite subset of X.

Proof. (1) =⇒ (2) Since X is compact, Xd is a compact closed discrete set of X. Hence Xd is finite. Thus{x} is closed, for each x ∈ Xd.

(2) =⇒ (1) Straightforward. �Remarks 2.3.

(1) If X is a compact T1-space, then X is an I-space if and only if Xd is finite.(2) Let X be a topological space and K(X) a compactification of X. Then Xd ⊆ K(X)d. And since {y} is

not open, for each y ∈ K(X) \X, K(X) \X ⊆ K(X)d.

Proposition 2.4. Let X be a topological space and K(X) a compactification of X. Then the following state-ments are equivalent.

(i) K(X) is an I-space;(ii) Xd ∪ (K(X) \X) is a c-finite subset of X.

Proof. (i) =⇒ (ii) By Remarks 2.3 (2), Xd ∪ (K(X) \X) ⊆ K(X)d. Hence Xd ∪ (K(X) \X) finite and {x}is a closed set of K(X), for each x ∈ Xd ∪ (K(X) \X) (Lemma 2.2).

(ii) =⇒ (i) Let x ∈ X \ Xd. Then {x} is an open set of X. Hence {x} is an open set of K(X), sinceK(X) \ X closed set of K(X). Thus K(X)d = Xd ∪ (K(X) \ X). So that K(X) is an I-space followsimmediately from Lemma 2.2. �

The following corollaries are an immediate consequence of Proposition 2.4.

Corollary 2.5. Let X be a topological space. Then the following statements are equivalent.

(1) The Alexandroff compactification of X is an I-space;(2) Xd is a c-finite subset of X.

Corollary 2.6. Let X be a T1-space. Then the following statements are equivalent.

(1) The Wallman compactification of X is an I-space;(2) Xd ∪ (wX\X) is a c-finite set of X.

3. Finite Herrlich compactification remainder

The characterization of space such that its compactification is an I-space is given using finite compact-ification remainder (see Section 2). This section deals with a characterization of space such its Herrlichcompactification is finite. First we recall the Herrlich compactification construction given by Herrlich in1993 [5]. Remark that in the case of T1-spaces the Herrlich compactification coincides with the Wallmancompactification. Let X be a T0-space and consider the set Γ (X) of all filters F on X that satisfy thefollowing two conditions:

(a) F does not converge in X.(b) Every finite open cover of X contains some member of F .

202 K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205

Let Ω(X) be the set of minimal elements of Γ (X) and define:

(i) X∗ω = X ∪Ω(X);

(ii) A∗ω = A ∪ {F | F ∈ Ω(X) and A ∈ F} for A ⊆ X.

Then βω = {A∗ω | A open in X} is a base for a topology T ∗

ω on X∗ω. Since (X∗

ω, T ∗ω ) is compact and X is

dense in X∗ω, the space with underlying set X∗

ω and topology T ∗ω will be called the Herrlich compactification

of X, and will be denoted by βωX.The following properties have been proved by Herrlich [5].

Properties 3.1. Let X be a T0-space. Then the following properties hold:

(1) Every F ∈ βωX\X has a base consisting of open sets.(2) (A ∩B)∗ω = A∗

ω ∩B∗ω for subsets A and B of X.

(3) Each finite subset of βωX\X is closed in βωX.

Remark 3.2. Let X be a T0-space, x ∈ X and F ∈ βωX\X. Since F does not converge in X, there existsan open neighborhood O of x such that O /∈ F . Hence O∗

ω is an open neighborhood of x in βωX such thatF /∈ O∗

ω. If, in addition, βωX\X is finite then x has an open neighborhood U∗ω such that F /∈ U∗

ω, for eachF ∈ βωX\X (in this case U∗

ω = U).

Lemma 3.3. Let X be a non-compact T0-space. Every compact closed set of X is a closed set of βωX.

Proof. Let F ∈ Ω(X) and F be a compact closed set of X. Since F does not converge to x, for each x ∈ F ,there exists an open neighborhood Ux of x such that Ux /∈ F . So F ⊆

⋃(Ux: x ∈ F ). Hence there exists

a finite subset G of F such that {Ux | x ∈ G} is a covering of F . Thus X \ F ∪ {Ux | x ∈ G} is a finitecovering of X. Then (X \F ) ∈ F , since Ux /∈ F , ∀x ∈ G. So (X \F )∗ω = (X \F )∪Ω(X). Therefore, F is aclosed set of βωX. �

For a topological space X, an open cover N of X is called a bad-covering (b-covering, for short) of X if ithas not a finite sub-cover. An open cover of X which is not b-covering, is called a good-covering (g-covering,for short) of X.

Lemma 3.4. Let X be a T0-space and O an open cover of X. Then following statements are equivalent:

(1) O is a g-covering of X;(2) For each F ∈ βωX\X, there exists O ∈ O such that O ∈ F .

Proof. (1) =⇒ (2) Let O be a g-covering of X, then there exists a finite subset O′ of O such that O′ isan open cover of X. Hence, for each F ∈ βωX\X, there exists O ∈ O′ such that O ∈ F . Thus for eachF ∈ βωX\X, there exists O ∈ O such that O ∈ F .

(2) =⇒ (1) Let O be an open cover of X such that, for each F ∈ βωX\X, there exists O ∈ O such thatO ∈ F . Hence O∗ = {O∗

ω | O ∈ O} is an open cover of βωX. Since βωX is compact, there exists a finitesubset O1 of O∗ such that O1 is an open cover of βωX. Thus {O | O∗

ω ∈ O1} is a finite sub-cover X. So Ois a g-covering of X. �Definition 3.5. Let X be a topological space, O an open b-covering of X and n ∈ N. Call O an n-b-coveringof X if there exists a collection of n open proper sets U of X such that O ∪ U is a g-covering of X, andO ∪ U\{U} is a b-covering, for each U ∈ U .

K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205 203

Now, we are in a position to give a characterization of spaces with finite Herrlich compactificationremainder.

Proposition 3.6. Let X be a T0-space and k ∈ N. Then the following statements are equivalent:

(1) Card(βωX\X) = k;(2) X has the following properties:

(i) For each b-covering O of X there exists n � k such that O is an n-b-covering.(ii) X has a k-b-covering.

Proof. (1) =⇒ (2) (i) Let O be a b-covering of X. By Lemma 3.4, there exists a collection H of n (withn � k) elements of βωX\X, such that F /∈ O∗

ω, for each F ∈ H and O ∈ O.Let U be a collection of n open sets of X such that for each element U of U there exists an unique element

F of H such that F ∈ U∗ω.

Since {V ∗ω | V ∈ O ∪ U} is an open cover of βωX and βωX is compact, O ∪ U is a g-covering of X. That

O ∪ U\{U} is a b-covering, for each U ∈ U is immediate by Lemma 3.4. So O is an n-b-covering.(ii) Since βωX\X is finite and each element of βωX\X does not converge in X, there exists, for each

x ∈ X, an open neighborhood Ox of x such that Ox /∈ F , for each F ∈ βωX\X. Set O = {Ox | x ∈ X}. Itis immediate that O is a b-covering of X (Lemma 3.4).

For each F ∈ βωX\X, there exists an open set UF of X such that UF ∈ F and UF /∈ G, for eachG ∈ βωX\(X ∪ {F}). Set U = {UF | F ∈ βωX\X}. So {A∗

ω | A ∈ O ∪ U} is an open cover of βωX. HenceO ∪ U is a g-covering of X, since βωX is compact. That O ∪ U\{U} is a b-covering is immediate fromLemma 3.4. Thus X has a k-b-covering.

(2) =⇒ (1) Let Γ be a subset of βωX\X such that Card(Γ ) = n. It is immediate that Γ is a closed setof βωX. Set O = {O open set of X such that O∗

ω ⊆ βωX\Γ}. By Lemma 3.4, O is a b-covering of X. Sincefor each F ∈ Γ , {F} is a closed set of βωX, there exists a collection V of k + 1 open sets of X such that:for each element V of V, V ∗

ω contains a unique element of Γ . By Lemma 3.4, O ∪ V is a g-covering of X,and O ∪ V\{V } is a b-covering, for each V ∈ V. Hence n � k (condition (i)). Thus Card(βωX\X) � k.

Using the hypotheses (ii), there exists O an open b-covering of X such that O is a k-b-covering of X.Let U be a set of k open sets of X such that O ∪U is a g-covering of X. Since, for each U ∈ U , O ∪U\{U}is a b-covering of X, there exists F ∈ βωX\X such that F ∈ U and F /∈ V , for each V ∈ U\{U}. Hencek � Card(βωX\X). Therefore k = Card(βωX\X). �Example 3.7. Let Z be the set of integers equipped with the following order: 2n � 2m (resp. 2n+1 � 2m+1)if and only if n � m. Equip Z with the left topology (that is, topology generated by {(↓ x) | x ∈ Z} with(↓ x) = {y | y � x}). Then Z is a non-compact T0-space. Set U = {2n | n ∈ Z} and V = {2n + 1 | n ∈ Z}.Then U and V are two open sets of Z. Hence, for each b-covering O of Z, O is a 1-b-covering or a 2-b-covering,since O ∪ {U, V } is a g-covering. Thus Card(βωZ \ Z) = 2.

Now, we are in a position to give a characterization of space such that its Herrlich compactification is anI-space.

Corollary 3.8. Let X be a T0-space. Then the following statements are equivalent.

(1) The Herrlich compactification of X is an I-space;(2) X has the following properties:

(i) Xd finite.(ii) There exists k ∈ N such that every b-covering of X is an n-b-covering with n � k, X has a

k-b-covering and X has not a k + 1-b-covering.

204 K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205

We give a characterization of spaces such that their Herrlich compactification is submaximal.

Proposition 3.9. Let X be a non-compact T0-space. Then the following statements are equivalent:

(1) The Herrlich compactification of X is submaximal;(2) X has the following properties:

(i) X is submaximal.(ii) Every dense set of X is co-compact.(iii) There exists k ∈ N such that every b-covering of X is a n-b-covering with n � k, X has a

k-b-covering and X has not a k + 1-b-covering.

Proof. Necessary condition. Straightforward.Sufficient condition. By Proposition 3.6, Card(βωX \ X) = k. Then X is an open set of βωX. Let D

be a dense set of βωX. Since D ∩ X is a dense set of X, D ∩ X is an open co-compact set of X; so thatX \ (D ∩X) is a closed set of βωX, by Lemma 3.3. On the other hand, there exist F1, . . . ,Fm ∈ βωX \X(m � k) such that βωX \D = (X \ (X ∩D)) ∪ {F1, . . . ,Fm}. Since {F1, . . . ,Fm} is a closed set of βωX,βωX \D is a closed set of βωX. Therefore βωX is submaximal. �

In order to characterize space such that its Herrlich compactification is nodec, we need the following newconcepts.

Definition 3.10. Let O be a collection of nonempty open sets of X and n ∈ N.

– Call O an n-open-collection if every open b-covering of X containing O, is an n-b-covering.– Call O a special-open-collection (sp-open-collection, for short) if Card(

⋂(X \ O: O ∈ O)) < +∞ and

for each x ∈⋂

(X \O: O ∈ O), {x} is not open.

Proposition 3.11. Let X be a T0-space. Then the following statements are equivalent:

(1) The Herrlich compactification of X is nodec;(2) X satisfies the following properties:

(i) X is s-nodec.(ii) For each sp-open-collection O, there exists n ∈ N such that O is an n-open-collection.

Proof. (1) =⇒ (2) That X is s-nodec follows from Lemma 1.7.Let O be an sp-open-collection and K =

⋂(βωX \ O∗

ω: O ∈ O). Then⋂

(X \ O: O ∈ O) = K ∩ X,Card(K ∩X) < +∞ and for each x ∈ K ∩X, {x} is not open. Hence K ∩X is a nowhere dense subset of X,since X is a T0-space. Thus K is a nowhere dense subset of βωX. So K is finite, since βωX is s-nodec.

Let k = Card(K ∩Ω(X)) and suppose that there exists a collection C of k + 1 open sets of X such thatfor every open b-covering N of X containing O, N ∪C is a g-covering of X, and N ∪C \{U} is a b-covering,for each U ∈ C.

Let U ∈ C. Since βωX = K∪(⋃

(O∗ω: O ∈ O)), there exists F ∈ K∩Ω(X) such that F ∈ U∗

ω and F /∈ V ∗ω ,

for each V ∈ C \ {U}. Hence Card(K ∩Ω(X)) � k + 1. Contradicting the fact that k = Card(K ∩Ω(X)).Thus there exists n � k such that O is an n-open-collection.

(2) =⇒ (1) Let N be a nowhere dense subset of βωX. Since clβωX(N) ∩X is also nowhere dense and X

is s-nodec, Card(clβωX(N) ∩ X) < +∞ and for each x ∈ clβωX(N) ∩ X, {x} is not open. So N ∩ X is anowhere dense subset of X and thus a compact closed set of X. Then N ∩X is a closed set of βωX. On theother hand, βωX \ clβωX(N) is open. Then there exists a collection of a nonempty open sets O of X such

K. Belaid, L. Dridi / Topology and its Applications 161 (2014) 196–205 205

that βωX \ clβωX(N) =⋃

(O∗ω: O ∈ O). Hence Card(

⋂(X \O: O ∈ O)) = Card(clβωX(N)∩X) < +∞ and

for each x ∈⋂

(X \ O: O ∈ O), {x} is not open. Thus O is an sp-open-collection. So there exists n ∈ N

such that O is an n-open-collection.Suppose that Card((βωX \X)∩ clβωX(N)) � n+1. Let A ⊆ (βωX \X)∩ clβωX(N) such that Card(A) =

n + 1. Since for every x ∈ clβωX(N) ∩ X and every F ∈ A, F does not converge to x, there exists anopen neighborhood Vx of x such that Vx /∈ F . Set V = {Vx | x ∈ clβωX(N) ∩ X}. So N = O ∪ V is ab-covering of X. Since O is an n-open-collection, there exists a collection C of n open sets of X such thatN ∪ C is a g-covering of βωX. Hence A ⊆ ∪(U∗

ω: U ∈ C). Since N ∪ C \ {U} is a b-covering of βωX,for each U ∈ C and by Lemma 3.4, Card(A) � n. This contradict the fact that Card(A) = n + 1. ThusCard(βωX \X) ∩N � Card((βωX \X) ∩ clβωX(N)) � n. So (βωX \X) ∩N is a closed set of βωX. ThenN is closed. Therefore βωX is a nodec space. �Example 3.12. Let Z be the set of integers equipped with the following order: 2n− 1 � 2n and 2n+ 1 � 2n,for each n ∈ [1, 4]. Let Z be equipped with the left topology. It is immediate that Z is s-nodec. Let Obe an sp-open-collection of Z. Then Card(

⋂(X \ O: O ∈ O)) = n < +∞ and {x} is not open, for each

x ∈⋂

(X \O: O ∈ O). Hence⋂

(X \O: O ∈ O) ⊆ {2, 4, 6, 8}. Thus O is 4-open-collection. That the Herrlichcompactification of Z is nodec follows immediately from Proposition 3.11.

Acknowledgement

The authors gratefully acknowledge helpful comments, and suggestions of the anonymous referee.

References

[1] A.V. Arhangel’skii, P.J. Collins, On submaximal spaces, Topol. Appl. 64 (1995) 219–241.[2] M.E. Adams, K. Belaid, L. Dridi, O. Echi, Submaximal and spectral spaces, Math. Proc. R. Ir. Acad. 108A (2) (2008)

137–147.[3] K. Belaid, L. Dridi, O. Echi, Submaximal space and door compactifications, Topol. Appl. 158 (2011) 1969–1975.[4] E. Bouacida, O. Echi, E. Salhi, Topologies associée á une relation binaire et relation binaire speciale, Boll. Unione Mat.

Ital. 7 (10-B) (1996) 417–439.[5] H. Herrlich, Compact T0-spaces and T0-compactifications, Appl. Categ. Struct. 1 (1993) 111–132.[6] M.M. Kovar, Which topological spaces have a weak reflection in compact spaces?, Comment. Math. Univ. Carol. 36 (3)

(1995) 529–536.[7] R. Levy, J.R. Porter, On two questions of Arhangel’skii and Collins regarding submaximal spaces, Topol. Proc. 21 (1966)

143–154.[8] W.J. Lewis, J. Ohm, The ordering of spec R, Can. J. Math. 28 (1976) 820–835.[9] J. Schröder, Some answers concerning submaximal spaces, Quest. Answ. Gen. Topol. 17 (1999) 221–225.

[10] H. Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938) 112–126.