Every Lindelof space (in particular, every [k.sub.[omega]]-space) and every metrizable space
is Dieudonne complete.
Dales  within the context of "function algebras" on a compact metrizable space
. In this framework, one proves that the intersections of peak sets are sets of the same type, while the peak points are dense in the Silov boundary; in effect, their set constitutes the unique minimal boundary called, Bishop boundary (ibid.).
Moreover, since Gr(H) is closed in U(H) we have that the infinite grassmannian Gr(H) is a completely metrizable space
This is equipped with the compact-open topology, and it becomes a compact metrizable space
(possibly infinite dimensional) with the following natural continuous C-action:
Let X be a metrizable space
and let d be a bounded metric on X.
A well-known result of Christensen [8, Theorem 3.3] asserts that a metrizable space
X is Polish if and only if X has a fundamental compact resolution.
As for the rest of topologies, combination of the following items imply any closed ball [B(H).sub.[parallel]*[parallel][less than or equal to]n] forms a second countable metrizable space
which leads us to conclude (B(H), [tau]) is again Lindelof.
Since every pseudocompact space is weakly pseudocompact and pseudocompact metrizable space
is compact, the following result follows from Theorem 3.7.
Let X be a non compact, locally compact and metrizable space
such that X = [union][K.sub.n] where [([K.sub.n]).sub.n] is an exhaustive sequence of compact subsets of X.
The following theorem for locally compact spaces and complete metrizable spaces
was proved in .
It follows that two compact metrizable spaces
[[OMEGA].sub.1], [[OMEGA].sub.2] are Baire isomorphic if and only if
Hence (T-spaces, stratifiable spaces, Moore spaces and metrizable spaces
are all D-spaces.