Asian Journal of Pure and Applied Mathematics

In this work, we introduce basic properties of the reals (otherwise the  \(x-a x i s)\) denoted by  \(\mathbb{R}\)and with members known as real numbers. The  major subsets of this  universal set are known to be  the  set natural  numbers  \(\mathbb{N}\) = {1,2,3,…}, the  set of all integers \(\mathbb{Z}\) = {…, -2,-1, 0, 1, 2…} and the sets of rational numbers (fractions)  \(Q=\left\{\frac{P}{q}, P, q \in \mathbb{Z}, q \neq 0\right\}\) .Again, we introduce the concept of extension in the topology of real line and hence attempt to generate a vital extension result which states that “A subset of is said to be extendable if and only if it is countable, among others’ To  achieves  this we  introduce  the popular well  ordering principle  with the aid  of which  we  established that for any continuous map on   any given countable set is  extendable. Hence we further try to apply the same principle to a collected union of   these extendable subsets in \(\mathbb{R}\) to see if the universal set \(\mathbb{R}\) is extendable. We also look at conditions guaranteeing the extendibility of any set or union of sets of real numbers.