We consider the
completeness of the following members of the Pervin quasi-proximity class of a
completely regular Hausdorff space: 𝒫ℱ, ℒℱ, 𝒮𝒞, ℱ𝒯 and ℱℐ𝒩ℰ. We show that
these completeness properties are extension properties, as defined hy R. G. Woods,
which for 𝒫ℱ, ℒℱ and 𝒮𝒞 are closely related to almost realcompactness. Indeed, in a
countably paracompact space of non-measurable cardinality, PF-completeness,
LF-completeness, SC-completeness and almost realcompactness coincide. We show
that the fine quasi-uniformity of any Σ-product with compact factors is almost
precompact, and it follows that no Σ-product is FINE-eomplete. If a Σ-product is
C∗-embedded in its Tychonoff product π, and if π is P-complete for any of the
completeness properties under consideration, then π is the maximal P-extension of
Σ.
