On some topological spaces that occur in functional analysis

This is a study of the topological properties of spaces of continuous real functions on compact sets in the topology of pointwise convergence. Compact subsets in these spaces are called functionally complete. The topological properties of functionally complete compacta are established, among them the fact that the density of each subspace of a functionally complete compactum is equal to its weight. Compacta of countable tightness having this last property are called exact. Each functionally complete compactum is exact. It is proved that each exact compactum is a Fréchet–Uryson space and satisfies the first axiom of countability on an everywhere dense set of points. The continuous image of an exact compactum is exact. Recently M. Vage has constructed a “naive” example of an exact but not functionally complete compact space. Another interesting question is: does there exist a non-metrizable, homogeneous, functionally complete compactum?