TY - JOUR

T1 - Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

AU - Fuchino, Sakaé

AU - Juhász, István

AU - Soukup, Lajos

AU - Szentmiklóssy, Zoltán

AU - Usuba, Toshimichi

N1 - Funding Information:
✩ The first author is supported by Grant-in-Aid for Scientific Research (C) No. 19540152 of the Ministry of Education, Culture, Sports, Science and Technology Japan. The first author would like to thank Joan Bagaria, Dmitri Shakhmatov, Frank Tall as well as members of Nagoya set-theory seminar for their valuable comments and suggestions. The second, third and fourth authors were supported by the Hungarian National Foundation for Scientific Research grant Nos. 61600 and 68262. The third author was partially supported by Grant-in-Aid for JSPS Fellows No. 98259 of the Ministry of Education, Science, Sports and Culture, Japan. The authors also would like to thank the anonymous referee for some helpful comments. * Corresponding author. E-mail addresses: fuchino@isc.chubu.ac.jp (S. Fuchino), juhasz@renyi.hu (I. Juhász), soukup@renyi.hu (L. Soukup), zoli@renyi.hu (Z. Szentmiklóssy), usuba@math.tohoku.ac.jp (T. Usuba).

PY - 2010/6

Y1 - 2010/6

N2 - We introduce a new reflection principle which we call " Fodor-type Reflection Principle" (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ≤א2.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ≤א1; are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ≤א1 are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).

AB - We introduce a new reflection principle which we call " Fodor-type Reflection Principle" (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ≤א2.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ≤א1; are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ≤א1 are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).

KW - Axiom R

KW - Locally compact

KW - Meta-Lindelöf

KW - Metrizable

KW - Reflection principle

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U2 - 10.1016/j.topol.2009.04.068

DO - 10.1016/j.topol.2009.04.068

M3 - Article

AN - SCOPUS:77951665240

SN - 0166-8641

VL - 157

SP - 1415

EP - 1429

JO - Topology and its Applications

JF - Topology and its Applications

IS - 8

ER -