$\pi H^{*}$-Closed Sets in Topological Spaces

Resumo

The study of generalized closed sets plays a central role in modern topology,
particularly in understanding finer variations of separation axioms and closure
operators. In this paper, we introduce and investigate the concept of
$\pi H^{*}$-closed sets, as a natural extension of $H^{*}$-closed, $\delta g$-closed,
$\pi g$-closed, and related structures. Several illustrative examples are
presented using discrete, indiscrete, and cofinite topologies to clarify the
distinctions and interrelations among these classes of sets. Fundamental
properties of $\pi H^{*}$-closed sets are established, showing that while finite
unions of $\pi H^{*}$-closed sets preserve the property, their finite intersections
may fail to do so. We further define the dual notion of $\pi H^{*}$-open sets and
derive equivalent characterizations in terms of $h$-closure and $h$-interior
operators. The study also introduces the class of $\pi H^{*}$-$T_{1/2}$ spaces,
where $\pi H^{*}$-closed sets coincide with $h$-closed sets, and provides several
equivalent formulations. Our results unify and extend earlier investigations on
generalized closed sets, offering a comprehensive framework for analyzing
closure-based generalizations and separation axioms. This contributes to a
deeper understanding of the structural richness of topological spaces and opens
new directions for further research in generalized topology.