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.
(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 |