# Projective versions of the properties in the Scheepers Diagram

## TL;DR

This paper investigates the projective versions of classical covering properties within the Scheepers Diagram, providing functional characterizations of these properties through continuous images in the context of topology.

## Contribution

It introduces and characterizes the projective versions of properties in the Scheepers Diagram using functional methods and continuous images.

## Key findings

- Characterization of projective properties via continuous images.
- Extension of classical covering properties to projective versions.
- Functional descriptions of these properties in topological spaces.

## Abstract

Let $\mathcal{P}$ be a topological property. A.V. Arhangel'skii calls $X$ projectively $\mathcal{P}$ if every second countable continuous image of $X$ is $\mathcal{P}$. Lj.D.R. Ko$\check{c}$inac characterized the classical covering properties of Menger, Rothberger, Hurewicz and Gerlits-Nagy in term of continuous images in $\mathbb{R}^{\omega}$. In this paper we study the functional characterizations of all projective versions of the selection properties in the Scheepers Diagram.