The various properties of
topological spaces in the classical Baire category theorems which imply the
Baire property also imply the stronger property of pseudo-completeness.
In contrast to some of these properties and to the Baire property, J. C.
Oxtoby has shown that pseudo-completeness is productive. The following
main result places pseudo-completeness in the context of linear topological
spaces: Let E and F be linear topological spaces and g be a continuous
linear mapping of E into F. If E is pseudo-complete, g is almost open, and
the completion of g[E] has a continuous metric, then g[E] is complete. The
proof of this result uses the difference theorem, but not an open mapping
theorem. The hypotheses lead to a discussion of conditions for a linear mapping
to be almost open and for a linear topological space to have a continuous
metric.
An example shows that, although a translation invariant continuous metric on a
linear topological space E extends to a translation invariant continuous
pseudo-metric on the completion of E, this extension need not be a metric,
even if it induces a normed topology on E. Other examples show that a
pseudo-complete linear topological space need not be complete in its natural
uniformity, and that the almost open condition of the main result may not be
omitted and is not implied by the combination of the other conditions and the
conclusion.
