Monotone countable paracompactness. (English) Zbl 0938.54026

Good, Chris; Knight, Robin; Stares, Ian

Topology Appl. 101, No. 3, 281-298 (2000).

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

Keywords:

monotone normality; \(\beta\)-space; wN-space; stratifiability

Citations:

Zbl 0242.54027