International Journal of Pure and Applied Mathematics 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 set with respect to a cluster topological space 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 ideal are given. AMS Subject Classification: 54A05, 54E52, 54G12 Key Words: cluster system, cluster topological space, ideal topological space, E-scattered set, E-nowhere dense set 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 Baire category theorem [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 closure, the interior 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 subspace topology 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.