Abstract
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).
| Original language | English |
|---|---|
| Pages (from-to) | 1415-1429 |
| Number of pages | 15 |
| Journal | Topology and its Applications |
| Volume | 157 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - 2010 Jun |
| Externally published | Yes |
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Keywords
- Axiom R
- Locally compact
- Meta-Lindelöf
- Metrizable
- Reflection principle
ASJC Scopus subject areas
- Geometry and Topology
Cite this
Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness. / Fuchino, Sakaé; Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán; Usuba, Toshimichi.
In: Topology and its Applications, Vol. 157, No. 8, 06.2010, p. 1415-1429.Research output: Contribution to journal › Article
}
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
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
UR - http://www.scopus.com/inward/record.url?scp=77951665240&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77951665240&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2009.04.068
DO - 10.1016/j.topol.2009.04.068
M3 - Article
AN - SCOPUS:77951665240
VL - 157
SP - 1415
EP - 1429
JO - Topology and its Applications
JF - Topology and its Applications
SN - 0166-8641
IS - 8
ER -