Available online xxxx MSC: 54A05 54D70 54E35Let P be a directed set and X a space. A collection C of subsets of X is P -locally finite if C = {C p : p ∈ P } where (i) if p ≤ p then C p ⊆ C p and (ii) each C p is locally finite. Then X is P -paracompact if every open cover has a P -locally finite open refinement. Further, X is P -metrizable if it has a (P × N)-locally finite base. This work provides the first detailed study of P -paracompact and P -metrizable spaces, particularly in the case when P is a K(M ), the set of all compact subsets of a separable metrizable space M ordered by set inclusion. Z. Feng et al. / Topology and its Applications 191 (2015) 97-118 is (i) N-metacompact, (ii) K(ω ω )-metacompact, or (iii) K(M )-metacompact, for some separable metrizable space M (respectively). The equivalence of (i) and (i) is due to Gruenhage [7], while the other two equivalences are due to Garcia, Orihuela and Oncina [3]. The first two authors gave in [2] a systematic and uniform development of the theory of those compact spaces X such that X 2 \ Δ is P -metacompact (P -Eberlein compact), giving alternative characterizations in terms of almost subbases, bases, networks and point networks.A key result from [2] is that a P -Eberlein compact space is Corson compact (embeds in a Σ-product of lines) if P has calibre (ω 1 , ω) (every uncountable subset of P contains an infinite subset with an upper bound), but if P is not calibre (ω 1 , ω) then every compact space of weight no more than ω 1 is P -Eberlein compact. This demonstrates the critical role that calibre (ω 1 , ω) plays. However Gul'ko (and so Talagrand and Eberlein) compacta have pleasant properties that general Corson compacta do not (for example, ccc Gul'ko compacta are metrizable in ZFC). The proofs use special properties of directed sets of the form K(M ). The authors would like to know what -if anything -is special about directed sets of the form K(M ) as compared to general directed sets with calibre (ω 1 , ω).Here we show (Theorems 47 and 49) that a pseudocompact space X such that X 2 \ Δ is P -paracompact for some P with calibre (ω 1 , ω) is metrizable. However, if P is not calibre (ω 1 , ω) then there is a compact non-metrizable space X such that X 2 \Δ is P -paracompact. This gives an 'optimal' extension of Gruenhage's result [7], that a compact space X with paracompact X 2 \ Δ is metrizable. We also show that a separable space is Lindelöf if P -paracompact, and metrizable if P -metrizable, for some P with calibre (ω 1 , ω).However developing further results about P -paracompact and P -metrizable spaces apparently needs P to be a K(M ). These results depend on alternative characterizations which do not (overtly) refer to a directed set. For example, we show (Theorem 50) that a space is K(M )-metrizable for some separable metrizable M if and only if it has a weakly σ-locally finite base (a base B = n B n where for each point x we have B = {B s : x is locally finite in B s }). With these in place we show that a first countable space is paracompact if ...
