On a modification of finite-dimensional Niemytzki spaces

In this paper we extend the construction of the Niemytzki plane to dimension $n\ge 3$. Further, we consider a poset of topologies on the closed $n$-dimensional Euclidean half-space similar to one from \cite{AAK} which is related to the Niemytzki plane topology. Then we explore properties of topologies from the poset.