Fodor-type Reflection Principle, metrizability and meta-Lindel¨ ofness
Saka´ e Fuchino, Istv´ an Juh´ asz, Lajos Soukup, Zolt´ an Szentmikl´ ossy and Toshimichi Usuba
Abstract
We introduce a new reflection principle which we call“Fodor-type Reflection Principle”(FRP). This principle follows from but strictly weaker than Fleissner’s Axiom R. So, for example, 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-Lindel¨of if all of its subspaces of cardinality ≤ ℵ1 are meta-Lindel¨of (Theorem 4.1). It follows from this theorem that, under FRP, every locally countably compact space X is metrizable if all of its subspaces of cardinality ≤ ℵ1 are metriz- able (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.
1 Introduction
In this note, we consider the following type of reflection property of a topo- logical spaceX. LetP be a property of a topological space andκa cardinal.
Date: December 1, 2008 (09:53 JST)
2000 Mathematical Subject Classification: 03E35, 03E65, 54D20, 54D45, 54E35 Keywords: Axiom R, locally compact, meta-Lindel¨of, metrizable
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 Foun- dation for Scientific Research grant no. 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.
(1.1) If a topological space X satisfies the property P, then there is a subspace of X of size < κ satisfying the property P.
By moving to the negation Q of P, (1.1) can be also seen as the transfer property of the from:
(1.2) If every subspace ofX of size < κsatisfies the property Q, thenX also satisfies Q.
The instance of (1.2), where κ is ℵ2 and Q “metrizable”, is studied extensively in the literature with the most prominent result in this context being the following theorem of Dow.
Definition 1.1. A topological space X is calledℵ1-metrizable if every sub- space of X of size ≤ ℵ1 is metrizable. More generally, X is said to be κ-metrizable (< κ-metrizable resp.) for a cardinal κ if every subspace of X of size ≤κ (< κ resp.) is metrizable.
Theorem 1.1. (A. Dow [6, Theorem 3.1]) Every countably compact ℵ1- metrizable space X is metrizable.
In particular, every compactℵ1-metrizable space is metrizable. Arhangel- skii [1] asked if every locally compact ℵ1-metrizable space is metrizable.
Balogh proved that the answer is positive under Fleissner’s Axiom R:
Theorem 1.2. (Z. Balogh [3, Theorem 2.2])Assume Axiom R. Then every locally compact ℵ1-metrizable space is metrizable.
Recall thatAxiom Ris the principle asserting that the following AR([κ]ℵ0) holds for all cardinals κ≥ ℵ2:
AR([κ]ℵ0) : For any stationary S ⊆ [κ]ℵ0 and ω1-club T ⊆ [κ]ℵ1, there is I ∈T such thatS∩[I]ℵ0 is stationary in [I]ℵ0.
Here, T ⊆[X]ℵ1 for an uncountable setX is said to beω1-club (or tight and unbounded in Fleissner’s terminology in Fleissner [9]) if
(1.3) T is cofinal in [X]ℵ1 with respect to⊆ and
(1.4) for any increasing chain⟨Iα : α < ω1⟩in T of lengthω1, we have
∪
α<ω1Iα∈T.
The assumption of Axiom R cannot be simply dropped from Theorem 1.2 since, under the existence of a non-reflecting stationary set, we obtain a very strong form of negative answer to the question of Arhangelskii as the next proposition shows. However, we shall prove in Section 4 that we can still weaken the assumption of Axiom R in the theorem to Fodor-type Reflection Principle which will be defined in Section 2 (see Definition 2.1) and that this principle is strictly weaker than Axiom R (see Section 3).
Given a topological spaceX and a familyF of open sets, let ord(x,F) =
|{F ∈ F : x ∈ F}| for x ∈ X and ord(F) = sup{ord(x,F) : x ∈ X}. We say that F is point countable if ord(F)≤ ℵ0.
Recall that a topological space X is said to be meta-Lindel¨of if every open cover B of X has a point countable refinement. It is clear that every paracompact space is meta-Lindel¨of. By the Stone theorem, every metriz- able space is paracompact.
Proposition 1.3. If there is a non-reflecting stationary set S ⊆ Eωκ for a regular cardinal κ ≥ ℵ2 then there is a non-meta-Lindel¨of (and hence non- metrizable), locally compact, locally countable<κ-metrizable spaceX of size κ.
Note that the usual subspace topology on such an S is non-metrizable and ℵ1-metrizable, but not locally compact.
Proof. Let I = {ξ+ 1 : ξ < κ}. The underlying set of X is the disjoint union S∪I. For each α ∈ S choose a countable subset aα ∈ [I ∩α]ℵ0 of order typeω which is cofinal inα. Now define the topology ofX as follows:
(1.5) the elements of I are isolated.
(1.6) a neighborhood base of α∈S is{{α} ∪(aα\β) : β < α}.
By the Fodor lemma, every open cover B of X has a point in X which is covered by κ many elements of B. It follows that X is not meta-Lindel¨of and, in particular, non-metrizable.
X is clearly locally compact, so it is also regular. By the definition of the topology on X, it is also clear that X is locally countable.
We show, by induction on δ, that, for all δ < κ, X ¹ δ has a σ-discrete base. Since κ is regular, this is enough by the Bing metrization theorem.
The only non-trivial step is when cf(δ) ≥ ω1. But then there is a club C ⊆δ with C∩(S∩δ) = ∅. Letγν, ν < λbe an increasing enumeration of C. Then X ¹ δ has the partition {X ¹ [γν, γν+1) : ν < λ} into clopen sets.
Each of the clopen sets has σ-discrete base by the induction hypothesis. So
X ¹ δ also does. (Proposition 1.3)
In Section 2, we introduce a new type of stationary reflection principle FRP (or FRP(κ) for regular cardinals κ ≥ ℵ1) to which the role of Axiom R in Balogh’s theorem can be localized. We show that FRP(κ) follows from RP([κ]ℵ0) (Theorem 2.5) where RP([κ]ℵ0) is a slight strengthening of what is called “Reflection Principle” and denoted by RP(κ) in Jech’s Millennium Book [13]. Since Axiom R implies RP([κ]ℵ0) for all cardinals κof cofinality
≥ω1, FRP is a consequence of Axiom R.
On the other hand, we show in Section 3 that it is consistent that FRP(κ) for all cardinalκ of cofinality ≥ω1 holds while RP([κ]ℵ0) does not hold for all κ≥ ℵ2 (Theorem 3.5, (1)).
In Section 4, we prove that the transfer property (1.2) holds for meta- Lindel¨ofness of locally compact spaces under FRP (Theorem 4.1). We show that the assertion of Balogh’s theorem (Theorem 1.2) follows from Theorem 4.1 (Corollary 4.4). Since FRP is strictly weaker than Axiom R, Corollary 4.4 is an essential improvement of Balogh’s theorem. Also, Theorem 3.5, (2) implies that these topological transfer theorems under FRP do not impose any restriction on the size of the continuum.
Since FRP(ω1) is simply equivalent to the Fodor lemma, we can easily single out the ZFC part of the proofs of these transfer theorems to obtain the corresponding ZFC results (Corollaries 4.5 and 4.6).
For a property Q, let us call a topological space almost Q if every sub- space Y of X of cardinality <|X| satisfies Q. In particular, X is almost metrizable if and only if X is<|X|-metrizable.
A natural variant of (1.2) would be:
(1.7) IfX is almost Q, then X satisfies Q.
For various propertiesQ, we can ask whether (1.7) holds for all topological spaces X in a given class C of topological spaces and consider this problem as a question on compactness of C (in the sense of abstract model theory) with respect to the property Q.
In Section 5, we present miscellaneous results concerning the metriz- ability (resp. meta-Lindel¨ofness) of almost metrizable (resp. almost meta- Lindel¨of) spaces X in various classesC of topological spaces.
Proposition 1.3 can be also seen as an anticompactness result:
Proposition 1.4 (A rephrasing of Proposition 1.3). If there is a non- reflecting stationary set S ⊆ Eωκ for a regular cardinal κ ≥ ℵ2 then there is a locally compact, locally countable space X of size κ which is almost metrizable but not meta-Lindel¨of (and hence not metrizable).
In Section 6, we show that the same kind of anticompactness of metriz- ability as Proposition 1.4 can also hold without the existence of non-reflecting stationary sets.
We tried to make this note easily accessible for both topologists and set-theorists, though some of the readers good in both fields might find our formulation a little bit overdetailed.
2 Fodor-type Reflection Principle
In this section, we introduce the principle which we call “Fodor-type Re- flection Principle (FRP)” and show that this principle follows from Axiom R. We show in the next section that this principle is strictly weaker than Axiom R and some other weakenings of it.
The applications of FRP on reflection properties of topological spaces mentioned in the introduction will be given in Section 4. Actually, it appears that most of the known applications of Axiom R are already provable under FRP (see also Fuchino, Sakai, Soukup and Usuba [12]).
Definition 2.1. Let κ be a cardinal of cofinality ≥ ω1. The Fodor-type Reflection Principle for κ (FRP(κ)) is the following statement:
FRP(κ) : For any stationary S ⊆Eωκ and mapping g :S →[κ]≤ℵ0 there is I ∈[κ]ℵ1 such that
(2.1) cf(I) = ω1;
(2.2) g(α)⊆I for all α∈I∩S;
(2.3) for any regressive f : S ∩I → κ such that f(α) ∈ g(α) for all α ∈ S ∩I, there is ξ∗ < κ such that f−1′′{ξ∗} is stationary in sup(I).
Note that, for S and I as above, S ∩I is stationary in sup(I). In particular, if S ∩ I were empty, then ∅ : S ∩ I → κ is a/the regressive function for which there is no ξ∗ as in (2.3).
Fact 2.1. FRP(ω1) holds in ZFC.
Indeed, if we take I =ω1 then the statement follows immediately from the Fodor Lemma.
Lemma 2.2. FRP(κ) fails for a singular κ.
Proof. Suppose that λ = cf(κ) < κ. Let ⟨αξ : ξ < λ⟩ be a continuously and strictly increasing sequence of ordinals in κ\λ cofinal in κ. Let S = {αξ : ξ ∈ Eωλ}. Then S ⊆Eωκ and S is stationary in κ. Let g : S → κ be defined by
(2.4) g(αξ) ={ξ} forξ ∈Eωκ.
Since S ∋ αξ 7→ ξ ∈ λ is regressive and strictly increasing, there cannot be any I ∈[κ]ℵ1 satisfying (2.3). This shows that FRP(κ) does not hold.
(Lemma 2.2) Definition 2.2. Let FRP be the asssertion: FRP(κ) holds for all regular κ≥ ℵ1.
For regular κ ≥ ℵ2, FRP(κ) is not provable in ZFC since, for example, the existence of a non-reflecting subset of Eωκ would refute FRP(κ). In section 6, we show that the non existence of non-reflecting subset of Eωκ even does not guarantee FRP(κ).
However we can show that FRP(κ) follows from RP([κ]ℵ0) (see Theorem 2.5).
Here, for a cardinal κ≥ ℵ2, RP([κ]ℵ0) is the following principle:
RP([κ]ℵ0) : For any stationary S ⊆[κ]ℵ0, there is an I ∈[κ]ℵ1 such that (2.5) ω1 ⊆I;
(2.6) cf(I) = ω1;
(2.7) S∩[I]ℵ0 is stationary in [I]ℵ0. The following is well-known:
Lemma 2.3. RP([κ]ℵ0)is equivalent to the assertion that for any stationary S ⊆ [κ]ℵ0, there are stationarily many I ∈ [κ]ℵ1 satisfying (2.5), (2.6) and (2.7).
AR([κ]ℵ0) implies RP([κ]ℵ0) for a cardinal κ of cofinality ≥ ω1 since T = {I ∈ [κ]ℵ0 : ω1 ⊆ I and cf(I) = ω1} is ω1-club. Jech [13] called a weakening of RP([κ]ℵ0) “Reflection Principle” which is obtained by drop- ping the condition (2.6) from the definition of RP([κ]ℵ0). Jech’s reflection
principle is sometimes also called “Weak Reflection Principle” in the liter- ature (see, e.g. K¨onig, Larson and Yoshinobu [14]) and so we denote this principle by WRP([κ]ℵ0).
Axiom R follows from MA+(σ-closed) (see Beaudoin [4]) which in turn is a consequence of Martin’s Maximum (see Foreman, Magidor and Shelah [10]). In more modern terminology of Foreman and Todorcevic [11], Axiom R is equivalent to the stationary reflection to a internally unbounded struc- ture (this fact is stated essentially in Dow [7] under the definition of Axiom R which is slightly stronger than the one we use here). Since MA+(σ-closed) is consistent with CH (modulo some large cardinal), all the reflection prin- ciples we treat here are compatible with CH.
It is still open if WRP([κ]ℵ0), RP([κ]ℵ0) and AR([κ]ℵ0) can be separated.
This seems to be a quite difficult problem if these principles should be ever separated: it is known that RP([ω2]ℵ0) and AR([ω2]ℵ0) are equivalent; under 2ℵ1 =ℵ2, WRP([ω2]ℵ0) and RP([ω2]ℵ0) are equivalent and, e.g. under GCH, WRP([ωn]ℵ0) and RP([ωn]ℵ0) for alln∈ωare equivalent (see K¨onig, Larson and Yoshinobu [14]).
Nevertheless, our Fodor-type Reflection Principle can be easily separated from these reflection principles (see the next section).
Let us begin with a useful characterization of FRP(κ):
Lemma 2.4. For a regular cardinal κ ≥ ℵ2, FRP(κ) is equivalent to the following FRP•(κ):
FRP•(κ): For any stationaryS ⊆Eωκ and mappingg :S →[κ]≤ℵ0 there is a continuously increasing sequence⟨Iξ : ξ < ω⟩of countable subsets of κ such that
(2.8) ⟨sup(Iξ) : ξ < ω1⟩ is strictly increasing;
(2.9) each Iξ is closed with respect to g and
(2.10) {ξ < ω1 : sup(Iξ) ∈ S and g(sup(Iξ))∩sup(Iξ) ⊆ Iξ} is stationary in ω1.
Proof.First, assume FRP(κ). LetS ⊆Eωκbe stationary andg :S →[κ]ℵ0. Without loss of generality, we may assume that g(α)∩α̸=∅for all α ∈S.
LetI ∈[κ]ℵ1 be as in the definition of FRP(κ) for theseS andg, and let
⟨Iξ : ξ < ω1⟩ be a filtration of I satisfying (2.8) and (2.9). This is possible by (2.1) and (2.2).
We show that ⟨Iξ : ξ < ω1⟩ satisfies (2.10) as well. Suppose not. Then {ξ < ω1 : sup(Iξ) ̸∈ S or g(sup(Iξ))∩sup(Iξ) ̸⊆ Iξ} includes a club set
⊆ω1. It follows that S∩I\S0 is non stationary in sup(I) for
S0 ={α ∈S∩I : α= sup(Iξ) for someξ < ω1 and g(α)∩α̸⊆Iξ}. Let f :S∩I →I be defined by
f(α) =
{min(g(α)∩α\Iξ) if α∈S0 and α= sup(Iξ);
min(g(α)) otherwise.
Then f is regressive andf(α)∈g(α) for all α∈S∩I. By the assumption, there is an α∗ ∈ I such that f−1′′{α∗} is stationary. In particular, S0 ∩ f−1′′{α∗} is stationary. Let ξ∗ ∈ω1 be such that α∗ ∈Iξ∗ and letβ ∈S0∩ f−1′′{α∗} be such thatβ >sup(Iξ∗). Let η < ω1 be such thatβ = sup(Iη).
Then α∗ ∈ Iξ∗ ⊆ Iη. Since β ∈ S0, we have f(β) ̸∈ Iη by the definition of f. It follows thatf(β)̸=α∗. This is a contradiction.
Now, assume FRP•(κ). Suppose that S⊆Eωκ is stationary and g :S → [κ]ℵ0. Let S0 ={α ∈S : α is closed with respect to g}. Since κ is regular S0 is still stationary. Let ⟨Iξ : ξ < ω1⟩ be as in the definition of FRP•(κ) forS0 andg ¹S0. LetI be the closure of∪
ξ<ω1Iξ∪{sup(Iξ) : ξ < ω1}with respect to g. By the definition of S0 and since sup(Iξ) ∈ S0 for stationary many ξ < ω1, we have sup(I) = sup(∪
ξ<ω1Iξ). In particular, {supIξ : ξ <
ω1} is a club subset of sup(I).
We claim that this I satisfies the conditions in the definition of FRP(κ).
It is clear that I satisfies (2.1) and (2.2). To see that it also satisfies (2.3), suppose that f :S∩I →κ is regressive and f(α)∈g(α) for all α∈S∩I.
By the assumption, {ξ ∈ ω1 : g(sup(Iξ))∩ sup(Iξ) ⊆ Iξ} is stationary.
Hence, letting S1 ={ξ ∈ω1 : f(sup(Iξ))∈Iξ}, we have that S1 ⊇ {ξ∈ω1 : g(sup(Iξ))∩sup(Iξ)⊆Iξ}
is also stationary. For each ξ∈S1, let h(ξ) = min{η < ω1 : f(sup(Iξ))∈Iη}.
Then the mapping h : S1 → ω1 is regressive. Thus, by the Fodor lemma, there is a stationary S2 ⊆ S1 such that h′′S2 = {η∗} for some η∗ ∈ ω1. Since Iη∗ is countable, there is a stationary S3 ⊆ S2 such that, for any ξ ∈S3,f(sup(Iξ)) =α∗ for some fixedα∗ ∈Iη∗. It follows thatf−1′′{α∗} ⊇ {sup(Iξ) : ξ∈S3} is stationary in sup(I). (Lemma 2.4)
Theorem 2.5. For any regular cardinalκ >ℵ1,RP([κ]ℵ0)impliesFRP(κ).
Proof.By Lemma 2.4, it is enough to show that RP([κ]ℵ0) implies FRP•(κ).
Suppose that S ⊆Eωκ is stationary and g :S →[κ]≤ℵ0. Let
(2.11) S0 ={a ∈[κ]ℵ0 : sup(a)∈S\a, g(sup(a))∩sup(a)⊆a}. Claim 2.5.1. S0 is a stationary subset of [κ]ℵ0.
⊢
Suppose that C ⊆ [κ]ℵ0 is a club. Let s : κ<ω → κ be such that C ⊇ C(s) = {a ∈ [κ]ℵ0 : s′′a<ω ⊆a} and let D= {α < κ : s′′α<ω ⊆ α}. Since κ is regular, D is a club subset of κ. So there is an α∗ ∈ S ∩D.Let ⟨αn : n ∈ ω⟩ be an increasing sequence of ordinals such that α∗ = supn∈ωαn. Let a∗ be the closure of a0 ={αn : n∈ ω} ∪(g(α∗)∩α∗) with respect to s. Since a0 is cofinal in α∗ and α∗ ∈ D, we have sup(a∗) = α∗. Thus a∗ ∈ S0. By the closedness of a∗ with respect to s, we also have
a∗ ∈C(s)⊆C.
⊣
(Claim 2.5.1)By RP([κ]ℵ0), there is I ∈[κ]ℵ1 such that (2.12) cf(I) = ω1;
(2.13) g(α)⊆I for all α∈I∩S;
(2.14) S0∩[I]ℵ0 is stationary in [I]ℵ0. Note that (2.13) is possible by Lemma 2.3.
Let ⟨Iξ : ξ < ω1⟩ be a continuously increasing sequence of countable sets with I = ∪
ξ<ω1Iξ such that ⟨sup(Iξ) : ξ < ω1⟩ is strictly increasing (this is possible by (2.12)).
Let
S1 ={ξ < ω1 : ξ is a limit andIξ ∈S0}and S2 ={ξ < ω1 : g(sup(Iξ))∩sup(Iξ)⊆Iξ)}.
By (2.14), S1 is a stationary subset of ω1. Now, by the definition (2.11) of S0, we have S2 ⊇S1. Thus S2 is stationary as well. (Theorem 2.5) Corollary 2.6. RP implies FRP. In particular, Axiom R implies FRP.
3 Separation of FRP from WRP
In this section, we prove the consistency of the Fodor-type Reflection Prin- ciple with the total negation of the Weak Reflection Principle.
The following lemma is well-known and easy to prove:
Lemma 3.1. For ℵ2 ≤κ≤κ′, if WRP([κ′]ℵ0) then WRP([κ]ℵ0).
For a proof of the following lemma, see e.g. Jech [13], Theorem 37.18.
Lemma 3.2 (S. Todorˇcevi´c). WRP([ℵ2]ℵ0) implies 2ℵ0 ≤ ℵ2.
Lemma 3.3 (S. Shelah). Suppose that P is a c.c.c. poset, S a stationary subset ofω1 andpα ∈Pforα∈S. Then, for all but countably many β ∈S, we have
pβ∥–P“{α ∈S : pα ∈G˙} is stationary in ω1”.
Proof. Suppose otherwise. Then we can construct a strictly increasing sequence ⟨αξ : ξ < ω1⟩ in S such that, for all ξ < ω1, there are qξ ≤P pαξ and club Cξ ⊆ω1 such that
(3.1) qξ∥–P“{α∈S : pα ∈G˙} ∩Cξ =∅”.
Let βξ∈ω1, ξ < ω1 be such that (3.2) βξ ∈S∩∩
{Cη : η < ξ}.
Claim 3.3.1. {qβξ : ξ < ω1} is an antichain.
⊢
For ξ < ξ′ < ω1, we have βξ′ ∈ Cξ by (3.2). Hence qβξ∥–P“pβξ′ ̸∈ G˙ ” by (3.1). Since qβξ′ ≤P pβξ′, it follows that qβξ∥–P“qβξ′ ̸∈ G˙ ”. Hence qβξand qβξ′ are incompatible.
⊣
(Claim 3.3.1)Now, by the c.c.c. of P, this is a contradiction. (Lemma 3.3) Theorem 3.4. Suppose that FRP(κ) holds and P is a c.c.c. poset. Then
∥–P“ FRP(κ) holds”.
Proof. Suppose that ˙S is a P-name of a stationary subset of κ and ˙g a P-name of a mapping from ˙S to [κ]ℵ0. Let
(3.3) S ={α ∈κ : p∥–P“α ∈S˙ for some p∈P”}.
Then S is a stationary subset ofκ. Let g :S →[κ]ℵ0 be defined by (3.4) g(α) = {β∈κ : p∥–P“β ∈g(α) for some˙ p∈P”}
for α∈S. g is well-defined by the c.c.c. of P.
By Lemma 2.4, there is a continuously increasing sequence ⟨Iξ : ξ < ω1⟩ with Iξ ∈[κ]ℵ0 for ξ < ω1 such that
(3.5) ⟨sup(Iξ) : ξ < ω1⟩is strictly increasing,
(3.6) Iξ is closed with respect to g for all ξ < ω1, and
(3.7) S1 ={ξ ∈ω1 : g(sup(Iξ))∩sup(Iξ)⊆Iξ} is stationary.
Forξ ∈S1, since sup(Iξ)∈S, there is apξ ∈Psuch thatpξ∥–P“ sup(Iξ)∈ S˙”. Hence, by Lemma 3.3, there is a ξ∗ ∈S1 such that
pξ∗∥–P“{ξ ∈S1 : pξ ∈G˙}is stationary in ω1”.
Let ˙S2 be a P-name of “{ξ ∈S1 : pξ ∈G˙}”. By the definition (3.4) of g (3.8) ∥–P“ ˙g(α)⊆g(α) for every α∈S˙”.
So, by the definition (3.7) of S1, we have
pξ∗∥–P“ ˙S2 ⊆ {ξ∈ω1 : ˙g(sup(Iξ))∩sup(Iξ)⊆Iξ}”.
By (3.8) and (3.6),
∥–P“ each Iξ, ξ < ω1, is closed with respect to ˙g”.
Thus pξ∗ forces that ⟨Iξ : ξ < ω1⟩ is as in the definition of FRP•(κ) for ˙S and ˙g.
Since the argument above can be repeated in P ¹ p for any p ∈ P, it
follows that ∥–P“ FRP(κ) ”. (Theorem 3.4)
Theorem 3.5. (1) Suppose that “ZFC + FRP” is consistent. Then so is “ZFC + FRP + ¬WRP([κ]ℵ0) for all κ≥ ℵ2”.
(2) If “ZFC + CH + FRP” is consistent, then “ZFC + FRP” is consistent with any value of the size of continuum possible under ZFC.
Proof. (1): Suppose that V |= “ZFC + FRP”. In V, let P = Cλ (= the Cohen forcing adding λ many Cohen reals) for some λ ≥ ℵ3. Then VP |= 2ℵ0 ≥ ℵ3. Hence, by Lemma 3.2 and Lemma 3.1, VP |= “¬WRP([κ]ℵ0) for allκ≥ ℵ2”. By Theorem 3.4,VP |= “FRP(κ) for all cardinalsκof cofinality
≥ω1”.
(2): Suppose that V |= “ZFC + CH + FRP”. In V, let λ be a car- dinal such that λℵ0 = λ. Then, for P = Cλ, we have VP |= 2ℵ0 = λ and
VP |=“FRP”. (Theorem 3.5)
It seems that we can only establish the consistency of FRP + ¬WRP under 2ℵ0 ≥ ℵ3 by the method as above. However, it is shown in Fuchino, Sakai, Soukup and Usuba [12] using a completely different method that FRP +¬WRP is also consistent with 2ℵ0 ≤ ℵ2 modulo some large cardinal.
4 Reflection property of meta-Lindel¨ ofness under FRP
Definition 4.1. We say that a topological spaceX issmall subspaces meta- Lindel¨of (ssmLfor short) if every subspace ofX of sizeℵ1 is meta-Lindel¨of.
Remembering the Definition 1.1 ofℵ1-metrizability, the natural wording for this notion might be “ℵ1-meta-Lindel¨of”. However “ℵ1-meta-Lindel¨of”
has been already used for a different notion in the literature and hence the present choice of the term “ssmL”.
Nevertheless, we shall also say for an uncountalble cardinal κ that a topological spaceX is< κ-meta-Lindel¨of (≤κ-meta-Lindel¨of resp.) if every subspace Y of X of cardinality < κ (≤κ resp.) is meta-Lindel¨of.
In the following, L(X) denotes the Lindel¨of number of the topological space X. That is,
L(X) = min{κ : for any open covering B of X, there is a subcoveringC ⊆ B of cardinality ≤κ} .
Theorem 4.1. (1) Assume that λ is an uncountable cardinal and, for each regular ω1 < κ ≤ λ, we have FRP(κ). Suppose that X is a locally separable countably tight space with L(X) ≤ λ. If X is ssmL, then X is meta-Lindel¨of.
(2) Assume FRP. Suppose that X is a locally separable countably tight space. If X is ssmL, then X is meta-Lindel¨of.
Proof. We shall prove only (1) since it is clear that (2) follows from (1).
IfX is locally separable, then every cover of X has a refinement consist- ing of separable subspaces of X. Thus it is enough to show the following (∗)κ for all κ≤λ.
(∗)κ For any locally separable, countably tight, ssmL spaceX, ifB is an open cover of X of cardinality κ consisting of separable subspaces of X, then B has a point countable refinement.
We prove (∗)κ by induction on κ. Let B ={Bα :α < κ} be an open cover of X as in (∗)κ. We want to find a point countable refinement ofB.
Case 1. κ=ℵ0.
B itself is a point countable cover ofX.
Case 2. κ is regular uncountable.
Let Gα =∪{Bβ :β < α} forα < κ, and S ={α < κ:Gα ̸=Gα}.
Claim 4.1.1. S is non-stationary.
⊢
We prove first the following weaker assertion:Subclaim 4.1.1.1. S∩Eωκ is non-staionary.
⊢
Suppose, toward a contradiction, thatS∩Eωκ were stationary. For each α∈S∩Eωκ let(4.1) pα ∈Gα\Gα.
Fix h :S∩Eωκ → κ such that pα ∈Bh(α). For each α < κ let Dα ∈[Bα]ℵ0 be dense in Bα. By pα ∈ ∪
β<αBβ = ∪
β<αDβ and since X is countably tight, there is g0(α)∈[α]ℵ0 such that pα ∈ ∪{Dβ :β ∈g0(α)}.
Let g(α) = g0(α)∪ {h(α)}.
Now apply FRP(κ) to these S∩Eωκ and g to obtain I ∈[κ]ℵ1 such that (4.2) cf(I) = ω1,
(4.3) h(α)∈I for eachα∈S∩Eωκ∩I, (4.4) g0(α)⊆I for each α∈S∩Eωκ∩I,
(4.5) for any regressive function f : S∩Eωκ∩I → κ with f(α) ∈ g0(α) for all α∈S∩Eωκ∩I, there is ξ∗ ∈I with supf−1{ξ∗}= supI.
Let Y ={pα :α∈S∩Eωκ∩I} ∪∪
{Dβ :β ∈I}.
Since |Y | =ℵ1,Y is meta-Lindel¨of. By (4.3) and (4.4), G ={Gα :α∈ I} covers Y. So it follows that G has a point countable refinement E. For each α∈S∩Eωκ∩I letEα ∈ E be such thatpα ∈Eα.
Since Eα is an open neighborhood of pα and pα ∈ ∪{Dβ :β ∈g0(α)} we have Eα ∩ ∪{Dβ : β ∈ g0(α)} ̸= ∅. Let f(α) ∈ g0(α) be such that Eα∩Df(α) ̸=∅.
By (4.5), there isξ∗ ∈I such thatJ =f−1{ξ∗}is unbounded inI. Since Dξ∗ is countable and Eα ∩Dξ∗ ̸= ∅ for all α ∈ J, it follows that there is d∈Dξ∗ such that K ={α ∈J :d∈Eα} is unbounded inI.
Since E is point countable, E′ = {E ∈ E : d ∈ E} is countable. Since Eα ∈ E′ for each α ∈ K, there are K′ ⊆ K and E∗ ∈ E such that K′ unbounded in I and Eα=E∗ for all α ∈K′.
Let β ∈ I be such that E∗ ⊆ Gβ. Then Eγ ∋ pγ ̸∈ Gβ for all γ ∈ (S∩Eωκ∩I)\β by (4.1). In particular, Eγ ̸=E∗ for any γ ∈K′\β. This is a contradiction to the choice of E∗ and K′.
⊣
(Subclaim 4.1.1.1) Let C be a club subset of κ consisting of limit ordinals such that S∩ Eωκ∩C =∅. LetD={α∈C : α\S is cofinal in α}.
Since D is a club subset of κ, we are done with the following subclaim.
Subclaim 4.1.1.2. S∩D=∅.
⊢
Let α ∈ D. If cf(α) = ω, then α ̸∈ S since D ⊆ C. So assume cf(α) > ω. Suppose that p ∈ Gα. By the countable tightness of X, there is an Y ∈ [Gα]ℵ0 such that p∈Y. By the definition of D there is a β < α such that Gβ ⊇ Y and β ∈ α\S. Then p ∈ Gβ = Gβ ⊆ Gα. This shows that Gα =Gα and hence α̸∈S.⊣
(Subclaim 4.1.1.2)⊣
(Claim 4.1.1) Let C be a club in κ\S consisting of limit ordinals and let ⟨γi : i < κ⟩ be an increasing enumeration of C. Let Hi =Gγi+1 \Gγi for i < κ. Then {Hi :i < ω1} is a partition ofX into clopen sets.Each Hi is covered by Ui ={Bξ\Gγi : γi < ξ < γi+1}. Since | Ui|< κ, Ui has a pointwise countable refinement Fi by the induction hypothesis.
Since Hi’s are pairwise disjoint, F =∪{Fi :i < κ} is also point countable.
Clearly F is an open covering of X refiningB. Case 3. κ is singular.
Let ⟨κi : i < cf(κ)⟩ be a continuously and strictly increasing cofinal sequence of cardinals in κ. Put Gi =∪
{Bα : α < κi} for each i <cf(κ).
By the induction hypothesis, for each i <cf(κ), there is a point count- able refinement Ci of the open covering {Bα : α < κi} of Gi. Note that each element C of Ci is separable since C is an open subset of some Bα, α < κi.
Put C =∪
i<cf(κ)Ci. Then C covers X and ord(C)≤cf(κ).
For C, C′ ∈ C, let C ∼ C′ if and only if C∩C′ ̸= ∅ and let ≈ be the transitive closure of ∼.
Claim 4.1.2. Each of the equivalence classes of ≈ has cardinality ≤cf(κ).
⊢
Let C ∈ C be arbitrary. Choose a countable dense subset D of C. If C ∼ C′ then there is d ∈ D ∩ C′. Since ord(d,C) ≤ cf(κ) for d ∈ D and D is countable, it follows that | {C′ ∈ C : C ∼ C′} | ≤ cf(κ). Hence| {C′ ∈ C : C ≈C′} | ≤cf(κ) as well.
⊣
(Claim 4.1.2) Let E be the set of all equivalence classes of the relation ≈. Then {∪e : e ∈ E} is a partition of X into disjoint open sets, and every ∪e iscovered bye. Since |e| ≤cf(κ)< κ, we can apply the induction hypothesis (∗)λ, λ < κ to get a point countable refinement Fe of e which cover ∪e.
Since e refines B for each e ∈ E, F = ∪
{Fe : e ∈ E} is as desired.
(Theorem 4.1) The statement of Theorem 1.2 can be slightly modified to obtain the following ZFC result:
Theorem 4.2. Suppose thatX is a locally compact meta-Lindel¨of space. If X is ℵ1-metrizable, then X is metrizable.
Proof.LetE be a point countable cover ofXconsisting of open subsets ofX with compact closures. By Dow’s theorem (Theorem 1.1), E is metrizable and hence separable for all E ∈ E. It follows that every E ∈ E is also metrizable and separable.
For E, E′ ∈ E, let E ∼ E′ if and only if E ∩E′ ̸= ∅. Let ≈ be the transitive closure of ∼.
Similarly to the proof of 4.1.2, we can show that each of the equivalence classes of ≈ has cardinality≤ ℵ0 because every E ∈ E is separable and the cover E is point countable.
Let E be the set of all equivalence classes of the relation ≈. Then {∪e:e∈E} is a partition ofX into disjoint open sets.
For e ∈ E, ∪e is a countable union of open subspaces which are metric spaces. So ∪e is also a metric space by the Bing metrization theorem.
ThusX can be partitioned into clopen subspaces which are all separable metric spaces. It follows that X is also metrizable. (Theorem 4.2)
The same argument as above also proves:
Proposition 4.3. Suppose thatX is a locally compact meta-Lindel¨of space.
If X is locally metrizable, then X is metrizable.
This proposition with the proof similar to the one above seems to be well-known.
Balogh’s theorem (Theorem 1.2) can be obtained now as a corollary of Theorems 4.1 and 4.2 under the Fodor-type Reflection Principle:
Corollary 4.4. (1) Let λ be a cardinal and assume that for each regular ω1 < κ≤λwe have FRP(κ). Suppose thatX is a locally countably compact space with L(X)≤λ. If X is ℵ1-metrizable, then X is metrizable.
(2) Assume FRP. Suppose thatX is a locally countably compact space.
If X is ℵ1-metrizable, then X is metrizable.
Proof. We prove only (1) since it is clear that (2) follows from (1).
Let X be as in (1). Then every point of X has a countably compact neighborhood, and this neighborhood is metrizable by Dow’s theorem (The- orem 1.1). Inparticular, the neighborhood is compact since countalbe com- pactness and compactness are equivalent for metrizable spaces. Thus the neighberhood is separable because compact metric spaces are separable. It follows also that X is countably tight.
X is ssmL since it is ℵ1-metrizable. Hence X is meta-Lindel¨of by The- orem 4.1 (1). By Theorem 4.2, it follows that X is metrizable.
(Corollary 4.4) As noted in Fact 2.1, FRP(ℵ1) is a theorem in ZFC. Thus the proofs of Theorem 4.1 and Corollary 4.4 also establish the following ZFC results:
Corollary 4.5. Suppose that X is a locally separable countably tight space with L(X)≤ ℵ1. If X is ssmL, then X is meta-Lindel¨of.
Corollary 4.6. Suppose that X is a locally compact space with L(X)≤ ℵ1. If X is ℵ1-metrizable, then X is metrizable.
5 Almost metrizability and almost meta-Lindel¨ ofness
The following theorem may be seen as a singular compactness theorem on meta-Lindel¨ofness of locally separable countably tight spaces analogous to the famous Shelah’s Singular Compactness Theorem on the notion of free- ness (Shelah [16]). This theorem shows that the regularity of κ in Proposi- tion 1.4 cannot be dropped.
Theorem 5.1. Suppose that X is a locally separable countably tight space and|X| is singular. IfX is almost meta-Lindel¨of then X is meta-Lindel¨of.
The proof of Theorem 5.1 will be given after Lemma 5.4.
Corollary 5.2. Suppose that X is a locally compact space and|X| is sin- gular. If X is almost metrizable then X is metrizable.
Proof. By Theorem 5.1 and Theorem 4.2. (See the argument of the proof
of Corollary 4.4.) (Corollary 5.2)
Proposition 5.3. Suppose thatX is a locally separable almost meta-Lindel¨of space. Then every open covering of X of cardinality < |X| consisting of separable subspaces has a point countable refinement.
Proof. The proof is quite similar to that of Theorem 4.1. It is enough to prove the following (∗)κ for all cardinal κ by induction on κ.
(∗)κ For any locally separable countably tight almost meta-Lindel¨of space X with |X| > κ, if B is an open cover of X of cardinality κ con- sisting of separable subspaces of X, then B has a point countable refinement.
Assume that (∗)κ′ holds for all κ′ < κ and let B = {Bα : α < κ} be an open cover of X as in (∗)κ.
The inductive proof of (∗)κ is divided into three cases just as in the proof of Theorem 4.1.
Case 1. κ≤ ℵ0.
B itself is a point countable cover ofX.
Case 2. κ is regular uncountable.
In this case, we have to prove the following Claim 5.3.1 corresponding to Claim 4.1.1 without FRP.
Let Gα =∪
{Bβ : β < α} forα < κ and S ={α < κ : Gα ̸=Gα}. Claim 5.3.1. S is non-stationary.
⊢
Toward a contradiction, suppose that S were stationary. For each α∈ S, let pα ∈ Gα\Gα. For α ∈κ, let Dα be a countable dense subset of Bα. Note that we have Gα =∪β<αDβ for α < κ.
Let A = {pα : α ∈ S} ∪∪
β<κDβ. Since X is almost meta-Lindel¨of and |A| ≤ κ < |X|, A as a subspace of X is meta-Lindel¨of. Thus there is a point countable open refinement E of the open covering {Gα : α < κ} of A. For each α ∈ S, choose Eα ∈ E such that pα ∈ Eα. Since Eα is an open neighborhood of pα and pα ∈ ∪
β<αDβ, there is f(α) < α such that Eα∩Df(α) ̸=∅.
By the (usual) Fodor lemma, there is a β∗ < κ such that T = {α ∈ S : f(α) =β∗} is stationary. Since Dβ∗ is countable, there is a d∗ ∈ Dβ∗
such that {α ∈ T : d∗ ∈ Dβ∗ ∩Eα} is unbounded in κ. Note that d∗ ∈ A by Dβ∗ ⊆ A. Hence, by point countability of E (on A), there is E∗ ∈ E such that d∗ ∈E∗ and {α ∈ T : E∗ =Eα} is unbounded in κ. Let γ < κ be such that E∗ ⊆ Gγ and let α ∈ κ\γ be such that E∗ = Eα. Then pα ∈Eα =E∗ ⊆Gγ. This is a contradiction to the choice of pα.
⊣
(Claim 5.3.1) The rest of this case is just as in Case 2 in the proof of Theorem 4.1.Case 3. κ is singular.
Let ⟨κi : i < cf(κ)⟩ be a continuously and strictly increasing cofinal sequence of cardinals in κ. Put Gi =∪
{Bα : α < κi} for i <cf(κ).
For i < cf(κ), if |Gi| = |X| there is a point countable refinement Ci
of the open covering {Bα : α < κi} of Gi by the induction hypothesis. If
|Gi| < |X|, there is also a a point countable refinement Ci of the open covering {Bα : α < κ}i of Gi by the almost meta-Lindel¨ofness ofX.
∪
i<cf(κ)Ciis then a point countable refinement ofB. (Proposition 5.3) Lemma 5.4. Suppose that X is a countably tight separable space. Then every dense set in X has a countable dense subset.
Proof. Fix a countable dense subset D∗ of X. For a given dense subset D of X, we want to find a dense countable D0 ⊆D. Since X is countably tight, for each p ∈ D∗ there is an Ap ∈ [D]ℵ0 such that p ∈ Ap. Then
∪
p∈D∗Ap is a countable dense subset of D. (Lemma 5.4) Proof of Theorem 5.1: We may assume that the underlying set ofX is a singular cardinal λ.
let us first show that, for every open covering B of X, there is an open covering B′ refining B with ord(B′)≤cf(λ).
For each α ∈ X, let Oα be an open neighborhood of α which is a separable subspace of X. Let Dα be a countable dense subset of Oα with
(5.1) α∈Dα.
Fix an increasing sequence of regular cardinals ⟨λi : i < cf(λ)⟩ cofinal in λ. For each i < cf(λ), let Pi = ∪
α<λiOα and Ei = ∪
α<λiDα. Then Pi is an open subset of X and Ei is a dense subset of Pi of size ≤ λi. By almost meta-Lindel¨ofness of X, each Ei is meta-Lindel¨of and thus there is a refinement Bi of {O∩Pi : O ∈ B} \ {∅} such that
(5.2) Ei ⊆∪ Bi
(5.3) ord(α,Bi)≤ ℵ0 for all α∈Ei. Note that we do not require Pi ⊆∪
Bi. However, since λi ⊆Ei by (5.1), it follows from (5.2) that B′ =∪
i<cf(λ)Bi is an open covering ofX refiningB. Thus, the next claim implies that B′ is as desired.
Claim 5.4.1. ord(α,Bi)≤ ℵ0 for every i <cf(λ) and α∈Pi.
