Monotone countable paracompactness.(English)Zbl 0938.54026

The authors define and study a monotone version of countable paracompactness. They opt to monotonize Ishikawa’s characterization and arrive at: a space is Monotonically Countably Paracompact if one can assign to each decreasing sequence $$F=\langle F_n\rangle_n$$ of closed sets with empty intersection a sequence $$U_F=\langle U_{F,n}\rangle_n$$ of open sets, with $$\bigcap_n \operatorname{cl}U_{F,n}=\emptyset$$, such that $$F_n\subseteq U_{F,n}$$ for all $$n$$ (this characterizes countable paracompactness) and if $$F_n\subseteq G_n$$ for all $$n$$ then $$U_{F,n}\subseteq U_{G,n}$$ for all $$n$$ (this is the monotonicity requirement) – demanding that only $$\bigcap_nU_{F,n}=\emptyset$$ leads to Monotone Countable Metacompactness.
MCP and MCM spaces turn out to be closely connected to the $$\beta$$- and wN-spaces of R. E. Hodel [Duke Math. J. 39, 253-263 (1972; Zbl 0242.54027)]; in fact MCM spaces are the same as $$\beta$$-spaces and every wN-space is MCP (the converse is true under mild assumptions, like local countable compactness). A consequence is the quite unexpected result that the familiar Sorgenfrey and Michael lines are not MCM; also the lexicographically ordered group $$\mathbb{Z}^{\omega_1}$$ is not MCP. Because $$X$$ is stratifiable iff the product $$X\times[0,1]$$ is monotonically normal there is no monotone parallel of Dowker’s characterization of normal countably paracompact spaces.
Reviewer: K.P.Hart (Delft)

MSC:

 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54E20 Stratifiable spaces, cosmic spaces, etc. 54E30 Moore spaces

Zbl 0242.54027
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