eu GENERALIZED NOWHERE DENSE SETS IN CLUSTER TOPOLOGICAL SETTING

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


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 continuation 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.
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 ⊂ 2 X \ {∅} 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 Remark 1.1.(see [1], [2], [3], [9]) Specially, if I is a proper ideal on X, then a cluster system The next definition introduces some basic notions derived from E-operator reminding the properties of local function which are known from an ideal topological space.
The family of all E-nowhere dense sets, locally E-scattered sets, nowhere dense sets, is denoted by N E , S E , 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 The following theorem summarizes the basic properties of N E .For completeness, we will prove it because some items are new or slightly different.
Theorem 2.1.(see [5]) (6): Let {H s } s∈S be a maximal family of pairwise disjoint open sets such that any H s is a subset of some set from {G t } t∈T and A := ∪ t∈T G t \ ∪ s∈S H s is nowhere dense.It is clear that B := ∪ s∈S H s is E-nowhere dense.By item (5), ∪ t∈T G t = A ∪ B is E-nowhere dense.The consequence follows from Lemma 2.2 (7). ( and Lemma 2.3.The opposite implication follows from the items (4) and ( 5). ( If E II = {E : E is of second category in (X, τ )}, then N E II is the family of all sets of first category.So, item (6) of Theorem 2.1 is a generalization of the Banach category theorem.

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 N E forms an ideal.
Theorem 3.1.Let E be a π-network in an open set G 0 .If A ⊂ G 0 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.
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 = {0, 1, 1 2 , 1 3 , . . ., 1 n , . . .} with the usual topology, E = {E : E is infinite}.It is clear that E is not a π-network in X.Put A = {1, 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 N E is an ideal.Theorem 3.2.(see [5]) If A is E-nowhere dense whenever A is E-nowhere dense, then N E is an ideal.
An obvious question is whether the assumption of Theorem 3.2 implies the equality N = N E and if the opposite implication holds.Next examples will give the negative answers.
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 = N E .Theorem 3.3.(see [5]) Let E be a π-network in X.Then the next conditions are equivalent: (1) A is E-nowhere dense if and only is A is E-nowhere dense, In [5] it is recommended to investigate a condition under which N E is an ideal.In this section we introduce a notion of additive cluster system.Theorem 3.4.N E is an ideal if and only if any sum of two locally Escattered sets is from N E .
Proof."⇒" By Theorem 2.1 (4), any locally E-scattered set is from N E , so the sum of two E-scattered sets is from N E ."⇐" Let A, B ∈ N E .By Theorem 2.1 (7) ) is a sum of a locally E-scattered and a nowhere dense set, so A ∪ B ∈ N E , by Theorem 2.1 (7).
Corollary 3.1.If S E is an ideal, then N E is so.
The opposite implication does not hold, as the next example shows.

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 ∈ N E iff E(A) = ∅.Next example shows that similar characterization exists for the ideal N of all nowhere dense sets.

Theorem 4 . 2 .
A ∈ N E if and only if E * (A) = ∅.Proof.If E * (A) = ∅, then A ∈ N E * and by Theorem 4.1, A ∈ N E .Let A ∈ N E and suppose E * (A) = ∅.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)) • = ∅.Since A ∈ N E , 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.
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 N E 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