Some Theorems on Compactness and Completeness

In this work, we prove the validity of the converses of some theorems about compactness and completeness. After we give some required basic definitions and theorems, we define monolimit property for sequences and nets, convergent subsequences property for first countable Hausdorff space, convergent subnets property for general Hausdorff space, and also, we show that those properties are equivalent to compactness and sequential compactness. On the other hand, we prove that a necessary and sufficient condition for completeness of a metric space is that every totally bounded subset of this space is relatively compact. Finally, we give some examples from some abstract spaces and normed spaces for application