Selection principles and proofs from the Book

TL;DR

This paper offers simplified proofs for key theorems in selection principles, clarifying their implications and introducing new results, thereby advancing understanding in the field of topology and set theory.

Contribution

It provides simplified proofs of fundamental theorems in selection principles and introduces new results related to these principles.

Findings

Simplified proofs of four key theorems in selection principles.

Completion of all provable implications in the Scheepers Diagram.

New results derived from the simplified proofs.

Abstract

I provide simplified proofs for each of the following fundamental theorems regarding selection principles: 1. The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space. 2. The Scheepers Diagram Last Theorem, due to Peng, completing all provable implications in the diagram. 3. The Menger Game Theorem, due to Telg\'arsky, determining when Bob has a winning strategy in the game version of Menger's covering property. 4. A lower bound on the additivity of Rothberger's covering property, due to Carlson. The simplified proofs lead to several new results.