Technical Observations on the δ-Sharp Operator: Support Representation, Topology, and Corrections to Selected Claims

This technical note revisits selected claims in the paper by A. Qahis and H. H. Al-Jarrah, “The δ-Sharp Operator in Ideal Topological Spaces and Its Induced Topology,” published in the Journal of Mathematics (2026), Article ID 7852731. We establish an exact support representation for the sharp and δ-sharp operators and derive an explicit characterization of the topology induced by the associated Kuratowski closure operator. These identities demonstrate that the topology inclusions discussed in the original article need not be strict and provide necessary and sufficient conditions for preservation of the whole space. We also present an explicit example clarifying the statement in Table 2, discuss the dependence of witnesses on neighborhoods in Corollary 4, and identify a minor wording issue in Example 2. Corrected arguments and mathematical explanations are supplied where appropriate. Importantly, these observations do not invalidate the principal Kuratowski closure construction or the established non-strict topology inclusions. The purpose of this note is to provide constructive mathematical clarification and additional structural results concerning the δ-sharp operator in ideal topological spaces.