On the Discrete Extension and Neighborhood Assignments

Let \( S \) be any proper subset of a topological space \( X \). We can introduce a finer topology \( \rho^{S} \) on \( X \) by designating all singletons in \( X \setminus S \) as open subsets. Each point in \( S \) retains the same open neighborhoods as in the original topology. We define a function \( \jmath \) as a neighborhood assignment or operator if it maps elements from \( X \) to the topology of \( X \), associates pairs of ordered disjoint closed subsets to the topology of \( X \), or links pairs \( (x, U) \), where \( U \) is an open neighborhood of \( x \), to the space \( X \). The space \( X \) is termed monotonically normal if there exists an \( M- \) operator on \( X \) that satisfies specific criteria. Our findings reveal that if \( X \) possesses the property of being monotonically normal, then for any proper subset \( S \) of \( X \), the discrete extension space \( X^{S} \) is also monotonically normal. Furthermore, we demonstrate that for a given topological space \( X \) and any finite subset \( S \subset X \), the discrete extension \( X^{S} \) achieves monotonic normality if either \( S \subset F_{1} \) or all elements of \( S \) lie outside \( F_{1} \) for every ordered pair \( (F_{1}, F_{2}) \) of disjoint closed subsets. Our exploration also examines the interplay between this type of extension and the concept of \( D- \) spaces. Notably, we establish that if \( X \) is a topological space and \( S \) is a compact proper subset of \( X \) such that \( X^{S} \) is discretely complete, then \( X^{S} \) qualifies as a \( D- \) space provided that \( X \setminus S \) is locally finite.