On the Review of Some Fundamental Classical Extension Results in the Set of Real Numbers | Asian Jou

We introduce basic characteristics of the reals (otherwise known as the denoted by and with members known as real numbers) in this paper. The sets of natural numbers = 1,2,3,..., the set of all integers =..., -2,-1, 0, 1, 2..., and the sets of rational numbers are the principal subsets of this universal set (fractions) We introduce the idea of extension in the topology of real lines once more, with the goal of generating a crucial extension result that states, "A subset of is said to be extensible if and only if it is countable, among others." To do this, we use the common well ordering concept to show that any continuous map on any countable set may be extended. As a result, we aim to apply the same idea to a collection of these extendable subsets in order to determine if the universal set is extensible. We also look at requirements that ensure that every set or union of sets of real numbers may be extended.

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