Abstract
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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
