Some Modifications and Improvements to Idealization of \(j\)-Approximation Spaces

Abstract

The core purpose of this paper is to point out some erroneous assertions given in the previous paper ”Idealization of \(j\)-Approximation Spaces”. In fact, we found some shortcomings and errors that need improvement and correction. Firstly, we prove that the relations given in the properties (6) and (7) in Proposition 4.1 are equal, not a subset or superset. Furthermore, we show that items (2), (3), (4), and (6) in Proposition 5.2 are incorrect in general. Additionally, counterexamples are given to confirm our claim. Finally, we illustrate that item (4) in Remarks 5.3 is also incorrect. Moreover, the correct form of the incorrect results in ”Idealization of \(j\)-Approximation Spaces” [22] is presented.