$C^*$-Embedded Dense Subsets of $z$-Neighborhood-Sublinear Spaces Are $P$-Embedded
Keywords:
generalized ordered space, globular space, lob-space, $P$-embedded, product, subspace of an ordinal, $z$-neighborhood-sublinear space, $C^*$-embeddedAbstract
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.
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