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The Arkhangel’skij-Tall problem under Martin’s axiom. (English) Zbl 0855.54006

There is a problem, namely, is every normal locally compact metacompact space paracompact? This problem and related ones have been considered under the axiomatic set theoretic axioms, for example, \(V = L\). In this paper, the authors obtain the following results under Martin’s axiom:
(1) \(\text{MA} (\omega_1) \to\) every normal locally compact meta-Lindelöf space is paracompact,
(2) \(\text{MA} \sigma\)-centered \((\omega_1) \to\) every normal locally compact metacompact space is paracompact.
Without any axiomatic set theoretic axioms, the following are obtained:
If a normal locally compact meta-Lindelöf space is \(\omega_1\)-collectionwise \(T_2\) space, it is paracompact. On the other hand, if there is a normal locally compact meta-Lindelöf space which is not paracompact, then there is one which is the union of \(\omega_1\)-many compact sets. All of the proofs are due to a general topology technique. There are minor misprints which will be easily found by the reader.
Reviewer: K.Iséki (Osaka)

MSC:

54A35 Consistency and independence results in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
03E35 Consistency and independence results
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