The strict topology for 𝑃-spaces

Abstract

A P P -space is a completely regular Hausdorff space X X in which every G δ {G_\delta } is open. It is shown that the generalized strict topologies β \beta and β 0 {\beta _0} coincide on C ∗ ( X ) {C^\ast }(X) , and that strong measuretheoretic properties hold; in particular, ( C ∗ ( X ) , β ) ({C^\ast }(X),\beta ) is always a strong Mackey space. As an application, an example is constructed of a non-quasi-complete locally convex space in which closed totally bounded sets are compact.