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The uniform order convergence structure on $\mathrm{ℳℒ}\left(X\right)$. (English) Zbl 1145.54003

If $X$ is a topological space and $u:X\to \overline{ℝ}$ is an extended real-valued function, one defines $I\left(u\right),S\left(u\right):X\to \overline{ℝ}$ by

$I\left(u\right)\left(x\right)=sup\left\{\text{inf}\left\{u\left(y\right):y\in U\right\}:U\in 𝒰\left(x\right)\right\}$

and

$S\left(u\right)\left(x\right)=\text{inf}\left\{sup\left\{u\left(y\right):y\in U\right\}:U\in 𝒰\left(x\right)\right\},$

where $𝒰\left(x\right)$ denotes the neighbourhood filter of $x$. The function $u$ is called normal lower-semicontinuous if $I\left(S\left(u\right)\right)=u$ and nearly finite if $\left\{x\in X:u\left(x\right)\in ℝ\right\}$ is open and dense in $X$. The set of all nearly finite, normal lower-semicontinuous functions on $X$ is denoted by $𝒩ℒ\left(X\right)$ and $ℳℒ\left(X\right)$ denotes the set of all $u\in 𝒩ℒ\left(X\right)$ 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 $𝒩ℒ\left(X\right)$. If $ℳℒ\left(X\right)$ carries the subspace uniform convergence structure, this structure induces the order convergence. He also proves that $𝒩ℒ\left(X\right)$ is the completion of $ℳℒ\left(X\right)$.

In the second part of the paper he uses the completion of the quotient of a suitable subspace to $ℳℒ\left(X\right)$ in order to construct generalized solutions of the Navier-Stokes euqations in three dimensions subject to an initial condition, which contain the classical ones.

##### MSC:
 54A20 Convergence in general topology 46E05 Lattices of continuous, differentiable or analytic functions 06F30 Order topologies (order-theoretic aspects) 35G25 Initial value problems for nonlinear higher-order PDE