$C^*$-Embedded Dense Subsets of $z$-Neighborhood-Sublinear Spaces Are $P$-Embedded - Hirata, Yasushi; Kemoto, Nobuyuki; Ohta, Haruto (Topology Proceedings, 2023)
We define a concept of $z$-neighborhood-sublinear space and point out that - every first-countable Tychonoff space and every generalized ordered space is $z$-neighborhood-sublinear, and - subspaces and finite products of $z$-neighborhood-sublinear spaces are $z$-neighborhood-sublinear. As a main theorem, we prove that every $C^*$-embedded dense subset of a $z$-neighborhood-sublinear space is $P$-embedded. The first author and Yukinobu Yajima, in $\left[C^*\right.$-embedding implies $P$-embedding in products of ordinals, Topology and Appl., 231 (2017), pp. 251-265], prove that for all subspaces $A$ and $B$ of an ordinal, if a closed subset $F$ of $A \times B$ is $C^*$-embedded in $A \times B$, then $F$ is $P$-embedded in $A \times B$. We can remove closedness from the assumption by applying the main theorem.