The locally fine coreflection and normal covers in the products of partition-complete spaces

We prove that the countable product of supercomplete spaces having a countable closed cover consisting of partition-complete subspaces is supercomplete with respect to its metric-fine coreflection. Thus, countable products of σ-partition-complete paracompact spaces are again paracompact. On the other hand, we show (Theorem 7.5) that in arbitrary products of partition-complete paracompact spaces, all normal covers can be obtained via the locally fine coreflection of the product of fine uniformities. These results extend those given in [K. Alster, Fund. Math. 114 (3) (1981) 173–181; L.M. Friedler et al., Pacific Math. J. 129 (2) (1987) 277–296; Z. Frolı́k, Bull. Acad. Pol. Sci. Math. 8 (1960) 747–750; A. Hohti and J. Pelant, Fund. Math. 136 (2) (1990) 115–120; A. Hohti and Z. Yun, Czechoslovak Math. J. 49 (124(3)) (1999) 569–583; J. Pelant, Czechoslovak Math. J. 37 (112) (1987) 181–187 and T. Plewe, Topology Appl. 74 (1996) 39–50].