Weak completeness of the Bourbaki quasi-uniformity
The concept of semicompleteness (weaker than half-completeness) is defined for the Bourbaki quasi-uniformity of the hyperspace of a quasi-uniform space. It is proved that the Bourbaki quasi-uniformity is semicomplete in the space of nonempty sets of a quasi-uniform space (X,U) if and only if each st...
| Main Author: | |
|---|---|
| Format: | info:eu-repo/semantics/article |
| Language: | English |
| Published: |
2017
|
| Online Access: | http://hdl.handle.net/10835/4862 https://doi.org/10.4995/agt.2001.3018 |
| Summary: | The concept of semicompleteness (weaker than half-completeness) is defined for the Bourbaki quasi-uniformity of the hyperspace of a quasi-uniform space. It is proved that the Bourbaki quasi-uniformity is semicomplete in the space of nonempty sets of a quasi-uniform space (X,U) if and only if each stable filter on (X,U*) has a cluster point in (X,U). As a consequence the space of nonempty sets of a quasi-pseudometric space is semicomplete if and only if the space itself is half-complete. It is also given a characterization of semicompleteness of the space of nonempty U*-compact sets of a quasi-uniform space (X,U) which extends the well known Zenor-Morita theorem. |
|---|