Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.56 (2024)

In this paper, using families of closed sets and a few modifications to the finite intersection features, we characterize the Rothberger Space and the Star-Rothberger Space. We also provide the selection principles that, in a reversed approach, can represent the Rothberger spaces and the star-Rothberger Spaces.

Abstract. Let An be the (unweighted) arithmetic mean of the first n prime numbers. We prove that for n ≥ 2, A 1+ α n log(n) n ≤ An+1 ≤ A 1+ β n log(n) n with the best possible constants α ≈ 0.43525 and β ≈ 1.22596. The right-hand side improves a result given by Z.-W. Sun in 2013.

Let X be an arbitrary set. Then a topology t on X is said to be completely useful if every upper semicontinuous linear (total) preorder ≾ on X can be represented by an upper semicontinuous real-valued order preserving function. In this paper, appealing, simple and new characterizations of completely useful topologies will be proved, therefore clarifying the structure of such topologies.

In this paper, we obtain some new inequalities for the confluent hypergeometric function. These results, at least for large argument, improve the well-known inequalities of Y. L. Luke and B. C. Carlson.