A Note on Monotone Countable Paracompactness

We show that a space is MCP (monotone countable paracompact) if and only if it has property (∗), introduced by Teng, Xia and Lin. The relationship between MCP and stratifiability is highlighted by a similar characterization of stratifiability. Using this result, we prove that MCP is preserved by both countably biquotient closed and peripherally countably compact closed mappings, from which it follows that both strongly Fréchet spaces and q-space closed images of MCP spaces are MCP. Some results on closed images of wN spaces are also noted. A space X is said to be monotonically countably metacompact (MCM) (see [1]) if there is an operator U assigning to each decreasing sequence (Dj)j∈ω of closed sets with empty intersection, a sequence of open sets U((Dj)) = ( U(n, (Dj)) ) n∈ω such that (1) Dn ⊆ U(n, (Dj)) for each n ∈ ω, (2) if Dn ⊆ En, then U(n, (Dj)) ⊆ U(n, (Ej)), (3) ⋂ n∈ω U(n, (Dj)) = ∅. X is said to be monotonically countably paracompact (MCP) if, in addition, (3′) ⋂ n∈ω U(n, (Dj)) = ∅. MCP spaces are precisely the monotonically cp of Pan [9]. Stratifiable spaces are MCP and semi-stratifiable spaces are MCM. MCM spaces are equivalent to β-spaces and MCP q-spaces coincide with wN-spaces, mirroring the relationship between stratifiable spaces and Nagata spaces, which are equivalent to stratifiable q-spaces. (Recall that a g-function on a space X with topology T is a mapping g : ω ×X → T such that x ∈ g(n, x) for a n ∈ ω. A space X is a q-space [8] if there is a g-function such that whenever xn ∈ g(n, x), the sequence (xn)n∈ω has a cluster point and is a wN-space [5] if, in addition, whenever g(n, x)∩ g(n, xn) 6= ∅, the sequence (xn)n∈ω has a cluster point). A space is said to have property (∗) if there is an operator V assigning to each closed set D a decreasing sequence V (D) = (Vn(D))n∈ω of open sets such that 1991 Mathematics Subject Classification. Primary: 54C10, 54D18, 54D20, 54E20, 54E30.