Ideal resolvability

The ideal generated by σ-nowhere dense sets

“…A strongly irresolvable space is a space all of whose open subspaces are irresolvable. The notion of Iresolvability, was introduced in [3], and is defined as follows: suppose I is an ideal on X, a topological space (X, τ ) is I -resolvable if there are two disjoint subsets A, B of X such that A * (I ) = B * (I ) = X. Observe that by definition, if a space is I -resolvable for some ideal I then it is resolvable, and if (X, τ ) is I -resolvable then it is J -resolvable for any ideal J ⊂ I .…”

“…It was shown in [3,Theorem 3.3] that a space is I m -resolvable if and only if it has two disjoint dense Baire subspaces. In the light of this result, it is interesting to consider the relationship between I σ -resolvability and the Volterra property.…”

“…The local function operator was used in [3] in the investigation of ideal resolvability. Observe that cl * (A) = A ∪ A * (I ) defines a Kuratowski closure operator, which generates a new topology τ * (I ) on X finer than τ .…”

Applications of Codense and Compatible Ideals

“…An ideal on a topological space as a non-empty collection of subsets of satisfying the following two conditions: (1) If and , then ; (2) If and , then . An ideal topological space [7] is a topological space with an ideal on , and is denoted by An ideal in an ideal topological space is called a codense ideal [8] if . Although Newcom [31], Jankovic and Hamlett [18] have used this as -boundary where as Dontchev [6] calls such spaces as Hayasi-Samuel spaces.…”

“…It is interesting that when [19]. In this ideal topological space a set is called dense [8] if , and the set is called -open if Again in ideal topological space, Modak and Bandyopadhyay [26,27] have proved two remarkable results:…”