SPACES MAKING CONTINUOUS CONVERGENCE AND LOCALLY UNIFORM-CONVERGENCE COINCIDE, THEIR VERY WEAK P-PROPERTY, AND THEIR TOPOLOGICAL BEHAVIOR

Although c = lu-spaces (namely, those defined by the title) were characterized internally some years ago by what can now be seen as the weakest in a range of P-properties, their general behaviour remained largely obscure. Our study of these properties casts some light on this, and enables us to place c = lu-spaces in their topological context. In addition to the obvious links with the P-property and with local compactness, some unexpected ones emerge as well, with countable box products and anti-compactness, for instance.