International Journal of Pure and Applied Volume 109 No. 2 2016, 459-467 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i2.19 ijpam.eu

GENERALIZED NOWHERE DENSE SETS IN CLUSTER TOPOLOGICAL SETTING

M. Matejdes Department of Mathematics and Computer Science Faculty of Education Trnava University in Trnava Priemyseln´a4, 918 43 Trnava, SLOVAKIA

Abstract: The aim of the article is to generalize the notion of nowhere dense with respect to a cluster which is defined as a triplet (X, τ, E) where (X, τ) is a topological space and E is a nonempty family of nonempty subsets of X. The notions of E-nowhere dense and locally E-scattered sets are introduced and the necessary and sufficient conditions under which the family of all E-nowhere dense sets is an are given.

AMS Subject Classification: 54A05, 54E52, 54G12 Key Words: cluster system, cluster topological space, ideal topological space, E-scattered set, E-nowhere

1. Introduction and Basic Definitions

Cluster topological spaces provide a general framework with the involvement of ideal topological spaces [1], [2], [3], [9]. They have a wider application and its progress can find in [6], [7], [10]. The paper can be considered as a con- tinuation of [5] where some cluster topological notions were introduced and it corresponds with the efforts to generalize the Baire classification of sets and the [4], [11]. In [5] one can find an open problem to discover a necessary and sufficient condition under which the family of all E-nowhere dense sets forms an ideal. In the first part we recall the basic notions and results of [5], a few counter examples are given and Section 3 is devoted to our main goal. Received: June 29, 2016 c 2016 Academic Publications, Ltd. Published: September 10, 2016 url: www.acadpubl.eu 460 M. Matejdes

In the sequel, (X, τ) is a nonempty topological space. By A, A◦, we denote ◦ the , the of A in X, respectively. By A we denote the interior of A. Definition 1.1. (see [5]) Any nonempty system E ⊂ 2X \{∅} will be called a cluster system in X. If G is a nonempty open set and any nonempty open subset of G contains a set from E, then E is called a π-network in G. For a cluster system E and a subset A of X, we define the set E(A) of all points x ∈ X such that for any neighborhood U of x, the intersection U ∩ A contains a set form E. A triplet (X, τ, E) is called a cluster topological space. If ∅= 6 Y ⊂ X and EY := {E ∩ Y : E ∈ E and E ∩ Y 6= ∅}, then (Y, τY , EY ) where τY is the is called a cluster topological subspace of (X, τ, E), provided EY 6= ∅. Remark 1.1. (see [1], [2], [3], [9]) Specially, if I is a proper ideal on X, then a cluster system EI = {E ⊂ X : E 6∈ I} leads to a local function of A, i.e., EI(A) = {x ∈ X : for any open set U containing x there is E ∈ EI such that E ⊂ U ∩ A} = {x ∈ X : U ∩ A 6∈ I, for any open set U containing x } =: A∗(I, τ) what is called the local function of A with respect to I and τ. Note, if E = 2X \ {∅}, then E(A) = A. The next definition introduces some basic notions derived from E-operator reminding the properties of local function which are known from an ideal topo- logical space. Definition 1.2. A set A is called E-scattered if A contains no set from E.A set A is locally E-scattered at a point x ∈ X if there is an open set U containing x such that U ∩ A is E-scattered (i.e., x 6∈ E(A)). A is locally E-scattered if A is locally E-scattered at any point from A (i.e., A ∩ E(A) = ∅) and A is E-dense in itself if A ⊂ E(A). A set A is E-nowhere dense if for any nonempty open set U there is a nonempty open subset H of U such that H ∩ A is E-scattered. The family of all E-nowhere dense sets, locally E-scattered sets, nowhere dense sets, is denoted by NE , SE , N , respectively. Remark 1.2. By Definition 1.1, E is a nonempty system of nonempty subsets of X. A trivial case E = ∅ (∅ ∈ E) leads to the trivial results, since X E(A) = ∅ and NE = 2 (E(A) = X and NE = ∅) for any A ⊂ X.

Definition 1.3. Let E1 and E2 be two cluster systems. E1 < E2 if for any E1 ∈ E1 there is E2 ∈ E2 such that E2 ⊂ E1. E1 and E2 are equivalent, E1 ∼ E2, if E1 < E2 and E2 < E1. GENERALIZED NOWHERE DENSE SETS... 461

2. Preliminary Results

The next properties of E-operator are clear and the proof of the following lemma is omitted. Lemma 2.1. (see [5]) (1) E(∅) = ∅, (2) E(A) is closed, (3) E(A) ⊂ A, (4) E(E(A)) ⊂ E(A), (5) E is a π-network in an open set G 6= ∅ if and only if E(G) = E(G) = G.

Lemma 2.2.

(1) If E1 ⊂ E2, then E1 < E2,

(2) if E1 < E2, then E1(A) ⊂ E2(A) and NE2 ⊂ NE1 ,

(3) if E1 ∼ E2, then E1(A) = E2(A) and NE2 = NE1 ,

(4) NE1∪ E 2 ⊂ NE i , i = 1, 2,

(5) NE i ⊂ NE1∩ E 2 , i = 1, 2,

(6) if A1 ⊂ A2, then E(A1) ⊂ E(A2),

(7) let A ⊂ Y ⊂ X. If A is EY -nowhere dense, then A ∈ NE ,

(8) if At is E-dense in itself for any t ∈ T , then ∪t∈T At is so, (9) if A is E-dense in itself, then A is so.

Proof. We will show only (7), (8) and (9). Other items are easy to prove. (7) Let G be a nonempty open subset of X. If G ∩ Y = ∅, there is noting to prove. Suppose G ∩ Y 6= ∅. Then G ∩ Y ∈ τY , so there is a nonempty open set H ∈ τ, such that ∅ 6= H ∩ Y ⊂ G ∩ Y and H ∩ Y ∩ A contains no set from EY , hence H ∩ Y ∩ A = H ∩ A contains no set from E. Since H ∩ G is a nonempty open subset of G and H ∩ G ∩ A contains no set from E, A ∈ NE . (8) It follows from ∪t∈T At ⊂ ∪t∈T E(At) ⊂ E(∪t∈T At). (9) Since A ⊂ E(A), A ⊂ E(A) = E(A) ⊂ E(A) by Lemma 2.1 (2) and Lemma 2.2 (6). Lemma 2.3. (see [5]) The next conditions are equivalent: 462 M. Matejdes

(1) A ∈ NE ,

(2) E(A) ∈ N ,

(3) (E(A))◦ = ∅.

The following theorem summarizes the basic properties of NE . For com- pleteness, we will prove it because some items are new or slightly different. Theorem 2.1. (see [5])

(1) N ⊂ NE . Consequently, if A ∈ N , then A ∈ NE ,

(2) if A ∈ NE and B ⊂ A, then B ∈ NE ,

(3) A \E(A) ∈ NE and A \E(A) is locally E-scattered,

(4) any E-scattered set is locally E-scattered and any locally E-scattered set is from NE ,

(5) if A ∈ N and B ∈ NE , then A ∪ B ∈ NE ,

(6) if Gt ∈ NE and Gt is open for any t ∈ T , then ∪t∈T Gt ∈ NE . Consequently,

if At ⊂ Y ⊂ X is τY -open and At ∈ NEY , then ∪t∈T At ∈ NE ,

(7) A ∈ NE if and only if A is a sum of a locally E-scattered set and a set from N ,

(8) if A, B ∈ NE and one of them is closed, then A ∪ B ∈ NE . ◦ Proof. (1): Let A ∈ N . By Lemma 2.1 (3), E(A) ⊂ A, so (E(A))◦ ⊂ A = ∅ and by Lemma 2.3, A ∈ NE . (2): It follows from the implications: B ⊂ A ⇒ E(B) ⊂ E(A) ⇒ (E(B))◦ ⊂ (E(A))◦ = ∅ and Lemma 2.3. (3): Let G be nonempty open. If G∩(A\E(A)) is empty, there is nothing to prove. Let x ∈ G ∩ (A \E(A)). Then there is an open subset H of G containing x, such that H ∩ A contains no set from E. Then H ∩ (A \E(A)) contains no set from E, so H ∩ (A \E(A)) is E-scattered. That means A \E(A) ∈ NE . The second part is clear. (4): The first part is clear. Let A be locally E-scattered. Since A∩E(A) = ∅, A = A \E(A) is E-nowhere dense by (3). (5): Let G be nonempty open. Since A is nowhere dense and B is E- nowhere dense, there are two nonempty open sets G0 ⊂ G and H ⊂ G0 such GENERALIZED NOWHERE DENSE SETS... 463

that A ∩ G0 = ∅ and B ∩ H is E-scattered. Hence (A ∪ B) ∩ H = B ∩ H is E-scattered, so A ∪ B is E-nowhere dense. (6): Let {Hs}s∈S be a maximal family of pairwise disjoint open sets such that any Hs is a subset of some set from {Gt}t∈T and A := ∪t∈T Gt \ ∪s∈SHs is nowhere dense. It is clear that B := ∪s∈SHs is E-nowhere dense. By item (5), ∪t∈T Gt = A ∪ B is E-nowhere dense. The consequence follows from Lemma 2.2 (7). (7): ” ⇒ ” It follows from equation A = (A \E(A)) ∪ (A ∩ E(A)), item (3) and Lemma 2.3. The opposite implication follows from the items (4) and (5). (8): Suppose A is closed. If A◦ = ∅, then A is nowhere dense and by item (5), A ∪ B ∈ NE . Let A◦ 6= ∅ and U be a nonempty open set. Suppose U ∩ A◦ = ∅. Since A \ A◦ is nowhere dense, so there is a nonempty open set H ⊂ U such that ◦ H ∩(A\A ) = H ∩A = ∅. Since B ∈ NE , there is a nonempty open set H0 ⊂ H, such that H0 ∩ B = (H0 ∩ A) ∪ (H0 ∩ B) = H0 ∩ (A ∪ B) contains no set from ◦ E, hence A ∪ B ∈ NE . Finally, suppose U ∩ A 6= ∅. Since A ∈ NE , there is a ◦ nonempty open set H1 ⊂ U ∩ A ⊂ A, such that H1 ∩ A = H1 contains no set from E, consequently (A ∪ B) ∩ H1 contains no set from E. So A ∪ B ∈ NE .

If EII = {E : E is of second category in (X, τ)}, then NEII is the family of all sets of first category. So, item (6) of Theorem 2.1 is a generalization of the Banach category theorem.

3. Main Results

Next theorem deals with a relationship between an E-nowhere dense set and a nowhere dense one and we will find some conditions under which NE forms an ideal.

Theorem 3.1. Let E be a π-network in an open set G0. If A ⊂ G0 is closed and E-nowhere dense, then A is nowhere dense. Consequently, if E(X) = X and A is a closed subset of X, then A is nowhere dense if and only if A is E-nowhere dense.

Proof. Let G1 be nonempty open. If A ∩ G1 = ∅, there is nothing to prove. Let A∩G1 6= ∅ and G := G1∩G0. Since E is a π-network in G0, H := G∩(X\A) 6= ∅ (if G ⊂ A, then there is a nonempty open subset H0 of G such that H0 ∩A = H0 contains no set from E, contradiction with assumption that E is a π-network in G0). So, H is a nonempty open subset of G and disjoint from A. 464 M. Matejdes

Remark 3.1. No assumption in Theorem 3.1 can be omitted. Let X = {a, b}, τ = {X, ∅}, E = {X}, A = {a}. Then E is a π-network in X. The set A is E-nowhere dense, A is not closed and A is not nowhere dense. The assumption that E is a π-network can not be omitted. Consider X = 1 1 1 {0, 1, 2 , 3 ,..., n ,... } with the usual topology, E = {E : E is infinite}. It is 1 clear that E is not a π-network in X. Put A = {1, 2 }. The set A is closed and E-nowhere dense. But A is not nowhere dense. It is known that A is nowhere dense iff A is so. An analogous equivalence for the E-nowhere dense sets leads to the fact that NE is an ideal. Theorem 3.2. (see [5]) If A is E-nowhere dense whenever A is E-nowhere dense, then NE is an ideal. An obvious question is whether the assumption of Theorem 3.2 implies the equality N = NE and if the opposite implication holds. Next examples will give the negative answers. X X Example 3.1. Let X = {a, b}, τ = 2 , E = {X}. Then NE = 2 and N = {∅}= 6 NE . Example 3.2. Let X = {a, b, c}, τ = {∅, {a, b},X}, E = {{a}}. Then NE = {∅, {b}, {c}, {b, c}} is an ideal. The set {b} ∈ NE , but {b} = {a, b, c} 6∈ NE . In the case if E is a π-network in X, the opposite implication is valid but the assumption that A is E-nowhere dense whenever A is E-nowhere dense seems to be too strong and it leads to the equation N = NE . Theorem 3.3. (see [5]) Let E be a π-network in X. Then the next condi- tions are equivalent:

(1) A is E-nowhere dense if and only is A is E-nowhere dense,

(2) N = NE .

In [5] it is recommended to investigate a condition under which NE is an ideal. In this section we introduce a notion of additive cluster system.

Theorem 3.4. NE is an ideal if and only if any sum of two locally E- scattered sets is from NE .

Proof. ”⇒” By Theorem 2.1 (4), any locally E-scattered set is from NE , so the sum of two E-scattered sets is from NE . GENERALIZED NOWHERE DENSE SETS... 465

”⇐” Let A, B ∈ NE . By Theorem 2.1 (7), A = A1 ∪ A2 and B = B1 ∪ B2, where A1,B1 are locally E-scattered and A2,B2 ∈ N . Then A ∪ B = (A1 ∪ B1) ∪ (A2 ∪ B2) is a sum of a locally E-scattered and a nowhere dense set, so A ∪ B ∈ NE , by Theorem 2.1 (7).

Corollary 3.1. If SE is an ideal, then NE is so. The opposite implication does not hold, as the next example shows. 1 1 1 Example 3.3. Let X = {0, 1, 2 , 3 ,..., n ,... } with the usual topology and 1 E = {E ⊂ X : X \ E is finite}. Then X1 = { n : n = 1, 2, 3,... } and X2 = {0} X are locally E-scattered, but X1 ∪ X2 = X is not so. It is clear NE = 2 is an ideal. Definition 3.1. A cluster system E is N -additive if for any A, B ⊂ X there is a nowhere dense set R, such that E(A ∪ B) = E(A) ∪ E(B) ∪ R.

Theorem 3.5. NE is an ideal if and only if E is N -additive.

◦ Proof. ”⇒” Let A, B ∈ NE . First, we will show (E(A ∪ B)) ⊂ E(A) ∪ E(B). Let x ∈ (E(A ∪ B))◦ and x 6∈ E(A) ∪ E(B). Then there is an open subset H ◦ of E(A ∪ B) containing x and H ∩ A and H ∩ B contain no set from E. So, H ∩ A and H ∩ B are E-nowhere dense set. Since NE is an ideal, there is a nonempty open subset G of H, such that G ∩ (A ∪ B) contains no set from E. On the other hand, x ∈ (E(A ∪ B))◦, hence G ∩ (A ∪ B) contains a set from E, a contradiction. Since E(A ∪ B) \ (E(A ∪ B))◦ in nowhere dense and E(A ∪ B) = [E(A ∪ B) \ (E(A ∪ B))◦] ∪ E(A) ∪ E(B), E is N -additive. ”⇐” Let A, B ∈ NE . Then E(A ∪ B) = R ∪ E(A) ∪ E(B), where R is a nowhere dense set. Since E(A), E(B) are nowhere dense, E(A ∪ B) is a nowhere dense set, so A ∪ B ∈ NE , by Lemma 2.3.

4. Derived Cluster Systems

It is well known that a set A is of first category if and only if D(A) = ∅ where D(A) is the set of all points in which A is of first category, i.e., for any x ∈ A there is an open set U containing x such that A ∩ U does not contain a set of second category. Question is if there is a similar characterization of E-nowhere dense sets, namely A ∈ NE iff E(A) = ∅. Next example shows that similar characterization exists for the ideal N of all nowhere dense sets. 466 M. Matejdes

◦ Example 4.1. Let EN = {E : E 6∈ N }. It is clear that EN (A) = A . By ◦ Lemma 2.3, A ∈ NE iff EN (A) is nowhere dense iff A is nowhere dense iff ◦ N A = ∅ iff A is nowhere dense. Consequently, NEN = N . So, A is a nowhere ◦ dense set iff A ∈ NEN iff EN (A) = A = ∅.

The next example shows that the equivalence A ∈ NE if and only if E(A) = ∅ does not hold in general. 1 1 1 Example 4.2. Let X = {0, 1, 2 , 3 ,..., n ,... } with the usual topology and E = {{1}} ∪ {E : E is infinite}. Then E(A) = ∅ iff A is finite and 1 6∈ A. It is clear NE = {A : 1 6∈ A}. ∗ Definition 4.1. Let E be a cluster system. Put E = {E : E 6∈ NE } = {E : (E(E))◦ 6= ∅} (see Lemma 2.3). A set A is called E-preopen if A ⊂ (E(A))◦.A cluster system of all nonempty E-preopen sets is denoted by Epo.

Now we will study a connection among NE , NE∗ and NEpo . Lemma 4.1. Let (E(A))◦ 6= ∅. If H is a nonempty open subset of (E(A))◦, then A ∩ H ∈ E∗ and A ∩ (E(A))◦ is E-preopen.

Proof. Denote G := (E(A))◦. First we prove H ⊂ E(A ∩ H). Let x ∈ H and U ⊂ H ⊂ G be an open set containing x. Since x ∈ U ⊂ G ⊂ E(A), A∩U = A∩H ∩U contains a set from E, so x ∈ E(A∩H). Since H ⊂ E(A∩H), (E(A ∩ H))◦ is nonempty, so A ∩ H ∈ E∗. Let x ∈ (E(A))◦. Then for any open set U containing x and U ⊂ (E(A))◦ ⊂ E(A) there is E ∈ E such that E ⊂ U ∩ A ⊂ U ∩ A ∩ (E(A))◦, hence x ∈ E(A ∩ (E(A))◦). We have proved (E(A))◦ ⊂ E(A ∩ (E(A))◦). That means A ∩ (E(A))◦ ⊂ (E(A))◦ ⊂ [E(A ∩ (E(A))◦)]◦, so A ∩ (E(A))◦ is E-preopen.

Theorem 4.1. Let E be a cluster system. Then Epo ∼ E∗ < E and NE = NE∗ = NEpo .

Proof. Since Epo ⊂ E∗, Epo < E∗. Let A ∈ E∗. Then (E(A))◦ 6= ∅ and by Lemma 4.1, A ∩ (E(A))◦ is E-preopen subset of A, so E∗ < Epo. That means ∗ po E ∼ E and by Lemma 2.2 (3), NE∗ = NEpo . ∗ The relation E < E is clear, so by Lemma 2.2 (2), NE ⊂ NE∗ . Let A ∈ NE∗ ◦ and A 6∈ NE . Then G := (E(A)) 6= ∅. Since A ∈ NE∗ , there is a nonempty open set H ⊂ G, such that A ∩ H contains no set from E∗. By Lemma 4.1, A ∩ H ∈ E∗, a contradiction.

∗ The following theorem gives a characterization of sets from NE by the E - operator. GENERALIZED NOWHERE DENSE SETS... 467

∗ Theorem 4.2. A ∈ NE if and only if E (A) = ∅.

∗ Proof. If E (A) = ∅, then A ∈ NE∗ and by Theorem 4.1, A ∈ NE . ∗ ∗ Let A ∈ NE and suppose E (A) 6= ∅. Let x ∈ E (A). Then for any open set G containing x the intersection G ∩ A contains a set B ∈ E∗ and by Definition ◦ ◦ 4.1, (E(B)) 6= ∅. Since A ∈ NE , for (E(B)) there is a nonempty open set H ⊂ (E(B))◦ such that A ∩ H does not contain a set from E. By Lemma 4.1 and Theorem 4.1, A ∩ H ∈ E∗ < E, so A ∩ H contains a set from E, a contradiction.

References

[1] J. Dontchev, M. Ganster, D. Rose, Ideal resolvability, Topology Appl., 93 (1999), 1–16. [2] E. Hatir, T. Noiri, On decompositions of continuity via idealization, Acta Math. Hungar., 96 (2002), 341–349. [3] E. Hatir, T. Noiri, On semi-I-open sets and semi-I-continuous functions, Acta Math. Hungar., 107 (2005), 345–353. [4] E. Korczak-Kubiak, A. Loranty, R. J. Pawlak, Baire generalized topological spaces, gen- eralized metric spaces and infinite games, Acta Math. Hungar., 140 (2013), 203–231. [5] M. Matejdes, On classification of sets in cluster topological space, Int. J. Pure and Appl. Math., 105, (2015), 269–280. [6] M. Matejdes, Generalized Volterra Spaces, Int. J. Pure and Appl. Math., 85 (2013), 955–963. [7] M. Matejdes, Characterization of Volterra Spaces, Int. J. Pure and Appl. Math., 88 (2013), 437–442. [8] V. Renukadevi, Relation between ideals and grills, Journal of Advanced Research in Pure Mathematics, 2 (2010), 9–14. [9] B. Roy, M. N. Mukherjee, On a typical topology induced by agrill, Soochow Journal of Math., 33 (2007), 771–786. [10] R. Thangamariappan, V. Renukadevi, Topology generated by cluster systems, Matemat- icki Vesnik, 67 No. 3, (2015), 174–184. [11] Zhaowen Li, Funing Lin, On I-Baire spaces, Filomat, 27 (2013), 301–310. 468