# Topological Nearly Entropy on Nearly Compact Spaces

## TL;DR

This paper explores properties of topological nearly entropy on nearly compact spaces, introduces a new entropy notion for nearly compact spaces, and examines their relationships and properties in product spaces.

## Contribution

It introduces a new topological nearly entropy for nearly compact spaces and studies its properties and relationships with existing notions.

## Key findings

- Topological nearly entropy is studied on nearly compact spaces.
- A new entropy notion, Ent_n(f), is introduced for nearly compact spaces.
- Fundamental properties of entropy in product spaces are established.

## Abstract

In our previous paper [9], we have introduced topological nearly entropy, Ent_N (f) by restricting X into a class of nearly compact spaces. In the present paper, some additional properties of this notion are studied. Furthermore, we introduce another new notion of topological nearly entropy of f denoted by Ent_n (f) when the whole space X itself is nearly compact. We show the relationship between these two notions for the class of nearly compact subspaces. We also propose new space, namely, R-space in studying the topological nearly entropy on nearly compact and Hausdorff space. As a consequence, the topological nearly entropy of f and it restriction f|K coincides. Finally, some fundamental properties of topological nearly entropy for product space are obtained.