Functional-Type Caristi–Kirk Theorem on Menger Space and Application

In this paper, we extend Caristi’s fixed point theorem in metric spaces to probabilistic metric spaces, and also, we prove some common fixed point theorems for a pair of mappings satisfying a system of Caristi-type contractions in the setting of a Menger space. Two examples are given to support the main results. Furthermore, we have functional equations as an application for the main theorem.

The Menger and projective Menger properties of function spaces with the set-open topology

For a Tychonoff space X and a family λ of subsets of X, we denote by Cλ(X) the space of all real-valued continuous functions on X with the set-open topology. A Menger space is a topological space in which for every sequence of open covers 𝓤1, 𝓤2, … of the space there are finite sets 𝓕1 ⊂ 𝓤1, 𝓕2 ⊂ 𝓤2, … such that family 𝓕1 ∪ 𝓕2 ∪ … covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space Cλ(X). Our main results state that Cλ(X) is Menger if and only if Cλ(X) is σ-compact; Cp(Y | X) is projective Menger if and only if Cp(Y | X) is σ-pseudocompact where Y is a dense subset of X.

On nearly Menger and nearly star-Menger spaces

In 1999, Kocinac defined and characterized the almost Menger property. Following this concept, we define and investigate nearly Menger and nearly star-Menger spaces. Every Menger space is nearly Menger, and every nearly Menger space is almost Menger. It is demonstrated that a nearly Menger space may not necessarily be a Menger space. In the similar way, we consider nearly ?-sets.