A pseudocomplete space is a topological space 
admitting a sequence
of families 
,
,
... of nonempty open sets such that every nonempty open set contains a member of each 
, and every choice 
satisfying
 |
for all 
has
 |
The bar denotes closure. This is Oxtoby's formulation of pseudocompleteness (Oxtoby 1961, Aarts and Lutzer 1973).
Every complete metric space is pseudocomplete. For example, take 
to be the nonempty open balls of radius less than 
. The nested closures
then have diameters tending to zero, and completeness
gives a point in every 
. A pseudocomplete regular space
is a Baire space (Aarts and Lutzer 1973).
Pseudocompleteness of countable closures in the real-valued continuous function space 
, with pointwise
convergence, is the hypothesis of Dow and Tkachuk's question about P-spaces
(Dow and Tkachuk 2025).
