| dc.description.abstract | Many studies have been conducted on dense topological subspaces over along period of time and interesting results have been obtained. Normality and compactness on topological subspaces have also been investigated by many mathematicians however, characterization when the subspaces are particularly dense has not been exhausted. In this study, we considered the case when the countable subspaces are dense. The objectives of the study are to: Characterize normality of dense topological subspaces and; characterize compactness of dense topological subspaces. The methodology involved description of both finite and infinite dimensional dense topological subspaces. One point-compactification and Tychonoff theorems have been used in the description of normality and compactness to prove situations where a dense topological subspace is countable. Concerning normality, results show that a topological subspace X is normal on every dense countable subset. Moreover, a subspace E of H is strongly normal in H if and only if E is normal in itself and for each continuous real-valued function f on E there exists a real-valued functioning on H continuous at all points of E which is an extension of f. Next, we have shown that if continuum hypothesis holds, then there is a countable dense set X of ℜc such that ℜc is normal on X. Furthermore, linearly ordered spaces are normal. On compactness, we have shown that if X is a compact space and Y is a Hausdorff space, then it implies that every continuous bijection f : X → Y is a homeomorphism. Also every locally compact Hausdorff space is Tychonoff. The results obtained are useful in explaining deformations and transformations in three dimensional objects. | en_US |