A STRONG COMPLETENESS CONDITION IN UNIFORM SPACES WITH WELL ORDERED BASES
A natural strengthening of the completeness hypothesis, called -completeness, allows us to extend to ! -metric spaces some properties of complete metric spaces which fail to hold if only the completeness condi- tion is assumed. In this way we achieve some results on hyperspace theory, dealing with supercompleteness and with selections of multivalued functions. The notion of -completeness is investigated and it is proved that quite important classes of ! -metric spaces have this property.
The Dugundji extension property can fail in $ω_μ$ -metrizable spaces
We show that there exist ωμ-metrizable spaces which do not have the Dugundji extension property (21 with the countable box topology is such a space). This answers a question posed by the second author in 1972, and shows that certain results of van Douwen and Borges are false.
The Invariance of the Vacuum is the Invariance of the World
The following theorem is demonstrated: If the vacuum is invariant under the group generated by the space integral of the time component of a local vector current, then the Hamiltonian is invariant also. A similar theorem holds for the group generated by the space integral of a space component of a local axial-vector current.