If is a topological space and is an extended real-valued function, one defines by
where denotes the neighbourhood filter of . The function is called normal lower-semicontinuous if and nearly finite if is open and dense in . The set of all nearly finite, normal lower-semicontinuous functions on is denoted by and denotes the set of all which are real-valued and continuous when restricted to the complement of a closed nowhere dense subset.
The author defines a uniform convergence structure on . If carries the subspace uniform convergence structure, this structure induces the order convergence. He also proves that is the completion of .
In the second part of the paper he uses the completion of the quotient of a suitable subspace to in order to construct generalized solutions of the Navier-Stokes euqations in three dimensions subject to an initial condition, which contain the classical ones.