On η-Local Functions in Ideal Topological Spaces

Abstract

This study introduces and investigates a new local function called \( \eta \)-local function in ideal topological space \( (X, \tau, I) \) by using the notion of \( \eta \)-open sets in topological space \( (X, \tau) \). The operator \( (\cdot)_\eta^* : \mathcal{P}(X) \rightarrow \mathcal{P}(X) \) is defined as \((\cdot)_\eta^* (A) = A_\eta^* = \big\{ x \in X \mid A \cap U \notin I \ \text{for every} \ U \in \eta\text{-}O(x) \big\}\) for each \( A \subseteq X \), where \( \eta \)-\( O(x) \) is the set of all \( \eta \)-open subsets of \( X \) containing \( x \). This study establishes some properties of \( A_\eta^* \) including its relationships to the local function and local function \( \Gamma^* \) in ideal topological space \( (X, \tau, I) \). This study also introduces a new type of closure called the \( \eta \)-local closure in ideal topological space \( (X, \tau, I) \) which is denoted by \( Cl_\eta^*(A) \) for each \( A \subseteq X \). Furthermore, this study establishes some properties of the \( \eta \)-local closure.