On Extendable Sets in the Reals (R) With Application to the Lyapunov Stability Comparison Principle of the Ordinary Differential Equations On Extendable Sets in the Reals (R) With Application to the Lyapunov Stability Comparison Principle of the Ordinary Differential Equations

Abstract

This work produces the authors’ own concept for the definition of extension on R alongside a basic result he tagged the basic extension fact for R. This was continued with the review of existing definitions and theorems on extension prominent among which are the Urysohn’s lemma and the Tietze extension theorem which we exhaustively discussed, and in conclusion, this was applied extensively in resolving proofs of some important results bordering on the comparison principle of Lyapunov stability theory in ordinary differential equation. To start this work, an introduction to the concept of real numbers was reviewed as a definition on which this work was founded.