NANO ALMOST I-OPENNESS AND NANO ALMOST I-CONTINUITY

Our purpose is to present the nano almost I -open and nano almost I -closed sets. Utilizing these new concepts the nano almost I -continuous functions have been obtained. We give a diagram that well illustrates the relations. AMS Mathematics Subject Classification: (2010) 54A05, 54A10 and 54B05.


Introduction
Topological spaces with ideals have been considered since 1930 by Kuratowski [1].The paper of Vaidyanathaswamy [2] in 1945 gave the subject great importance."A non-empty collection of subsets of X with heredity and finite additivity conditions is called as an ideal or a dual filter on X. Namely a non-empty family I ⊆ P (X)(P (X) is the set of all subsets of X) is nammed an ideal if and only if: i) A ∈ I gives P (A) ⊆ I (heredity).ii) A, B ∈ I gives A ∪ B ∈ I (finite additivity).Given X carries topology τ with an ideal I on X, a set operator() * : P (X) → P (X), named a local function [2] of A with respect to τ and I is defined as follows: for A ⊆ X, A * (I, τ ) = {x ∈ X : G x ∩ A / ∈ I for every G x ∈ τ (x)} where τ (x) = {G ∈ τ : x ∈ G}.A Kuratowski closure operator Cl * () for a topology τ * (I, τ ), named the * -topology finer than τ is defined by Cl * (A) = A ∪ A * (I, τ ) [2].When there is no chance for confusion, we will simply write A * for A * (I, τ ) and τ * for τ * (I, τ ).If I is an ideal on X, then the space (X, τ, I) is called an ideal topological space".
"The concept of nano topology was introduced by Lellis Thivagar and Carmel Richard [3] which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it and also they defined nano closed sets, nono interior and nano-closure".The concept of nano ideal topological spaces was introduced by Parimala et al. [4] and studied its properties and characterizations.The basic object of this paper is to present the nano almost I-open and nano almost I-closed sets.Utilizing these new concepts the nano almost I-continuous functions have been obtained.Nano almost I-openness and nano almost I-continuity are considered as a generalization of nano I-openness and nano I-continuity which are known before.Numerous nano topological properties of these new notions have been discussed.

Preliminaries
Before entering our working, we have compiled some basic facts on rough sets.Definition 2.1."(see [5]) Let U be a non-empty finite set of objects named the cosmos and R be an equivalence relation on U named as indiscernibility relation.Then U is divided into disjoint equivalence classes.Elements belonging to the some equivalence class are said to be indiscernible with one another.The pair (U, R) is said to be the approximation space.The lower approximation of X with respect to R is denoted by apr(X).That is, apr(X) = ∪{R(x) : R(x) ⊆ X; x ∈ U }, The upper approximation of X with respect to R is denoted by apr(X).That is, apr(X) = ∪{R(x) : R(x) ∩ X = φ; x ∈ U } and the boundary region of X with respect to R is denoted by B R (X).That is, B R (X) = apr(X) − apr(X) as Figure 1".(xi) apr[apr(X)] = apr[apr(X)] = apr(X)".Definition 2.3."(see [3]) Let U be the cosmos, R be an equivalence relation on U and τ R (X) = {U, φ, apr(X), apr(X), B R (X)}, where X ⊆ U .Then by Proposition 2.2, τ R (X) satisfies the condition of topology on U .That is, τ R (X) is a topology on U called the nano topology on U with respect to X and the pair (U, τ R (X)) is called a nono topological space.The elements of τ R (X) are called nano open sets in U and the complement of a nano open set is called a nano closed set.Elements of [τ R (X)] c being called duel nano topology of τ R (X)".Remark 2.4."Let (U, τ R (X)) be a nano topological space with respect to X where X ⊆ U and R be an equivalence relation on U .Then U/R denotes the family of equivalence classes of U by R".Definition 2.5."Let ([3], Definition 3.1) (U, τ R (X)) be a nano topological space and A ⊆ U .Then A is said to be:

Nano Ideal Topological Spaces
Recently in 2016, Thivagar and Devi [8] have considered the nano local function in nano ideal topological space and they have obtained a new topology.Before starting the discussion we shall consider the following concepts.
} is named the nano local function of A with respect to I and τ R (X).We will simply write ) be a nano topological space with an ideal I on U and for every ) be a nano topological space with ideals I, J on U and A, B be subsets of U .Then the following statements are true: The converse implications of (i), (ii) and (iii) of Theorem 3.3 do not hold in general, as seen from the next instance Proof.It is clear from Definition 3.6 and Theorem 3.4.

Nano Almost I-open Sets
The fourth section we have interpreted the properties of nano almost I-open sets in terms of its approximations.
Definition 4.1.In a nano ideal topological space (U, τ R (X), I), When there is no chance of confusion, the collection of all nano almost I-open sets of (U, τ R (X), I) will be symbolized by N AIO(U, τ R (X)).Also, N AIO(U, x) means the class of all nano almost I-open sets containing x ∈ U ." Recall that, a subset A of a nano ideal topological space (U, τ R (X), Proof.Let (U, τ R (X), I) be any nano ideal topological space and W i ∈ N AIO(U, τ R (X)) for i ∈ ∇.This means that for each i ∈ ∇,  The connections between nano almost I-openness with some other corresponding types have been given throughout the following implication in Figure 2.
Proof.By assumption and the fact that . Hence the result.
Proposition 4.9.The following statements are hold.
The next theorem gives several characterizations of nano almost I-continuous functions.

Proposition 4 . 2 .
Arbitrary union of nano almost I-open sets is also nano almost I-open.

Remark 4 . 3 .
A finite intersection of nano almost I-open sets need not in general be nano almost I-open as shown in the next example.

Figure 2 :
Figure 2: Relationship between some forms of near nano open sets.

Figure 3 :
Figure 3: Relationship between some weak forms of nano continuity.