Zero-dimensional spaces from linear structures
This paper contains some interesting results concerning topological spaces which are stratifiable over regular infinite cardinals \(\omega_\mu\). These were introduced by the author in [Pac. J. Math. 43, 253-266 (1972; Zbl 0226.54027)]. Example 1. There exists an \(\omega_\mu\)-Nagata space that does not have an ortho-base. In particular, for \(\omega_\mu= \omega_0\), there exists a Nagata space that does not have an ortho-base. Example 2. For every \(\omega_\mu\) there exists an \(\omega_\mu\)-stratifiable space that is not \(\kappa\)-Nagata for any cardinal \(\kappa\). In particular, there exists such a space that is stratifiable. Example 3. There exists a stratifiable space \(X\) such that the box product \(\square X^{\omega_0}\) is stratifiable over \(\omega_1\) (hence hereditarily paracompact and monotonically normal) but not stratifiable. Example 4. For \(\mu> 0\) there exists a space \(X\) which is stratifiable over \(\omega_\mu\) (and no other regular cardinal) such that \(\omega= \psi(X)< \omega_\mu< \omega_{\mu+1}\leq \chi(X)\), where \(\psi(X)\) is the pseudocharacter and \(\chi(X)\) is the character of \(X\). Example 5. For every \(\omega_\mu\) there exists an \(\omega_\mu\)-stratifiable space \(X\) such that \(|X|= \omega_\mu\) and every point of \(X\) has character greater than \(\omega_\mu\). Theorem. Every \(\omega_1\)-additive paracompact space is ultraparacompact. In particular, every \(\omega_1\)-additive linearly stratifiable space is ultraparacompact.
scientific article; zbMATH DE number 1893622
scientific article