On the Review of Some Fundamental Classical Extension Results in the Set of Real Numbers
Eziokwu, C. Emmanuel
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
Nnubia Agatha
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
Okereke, N. Roseline
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Immunogenetics, Subset of real numbers, H-Y and H-2 antibodies, countable sets, sex ratio, compact subsets, extendable sets, normal space
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