© Instytut Matematyczny PAN, 2017. By analyzing Dow's construction, we introduce a general construction of regular Lindelof spaces with points Gδ. Using this construction, we prove the following: Suppose that either (1) there exists a regular Lindel of P-space of pseudocharacter ≤ ω1 and of size > 2ω, (2) CH and (ω2) hold, or (3) CH holds and there exists a Kurepa tree. Then there exists a regular Lindel of space with points Gδ and of size > 2ω. This shows that, under CH, the non-existence of such a Lindel of space has a large cardinal strength. We also prove that every c.c.c. forcing adding a new real creates a regular Lindel of space with points Gδ and of size at least (2ω1 )V.

It is known that in X=A×B, where A and B are subspaces of ordinals, all closed C⁎-embedded subspaces of X are P-embedded. Also it is asked whether all closed C⁎-embedded subspaces of X are P-embedded whenever X is a subspace of products of two ordinals. In this paper, we prove that both of the following are consistent with ZFC: • there is a subspace X of (ω+1)×ω1 such that the closed subspace X∩({ω}×ω1) is C⁎-embedded in X but not P-embedded in X, • for every subspace X of (ω+1)×ω1, if the closed subspace X∩({ω}×ω1) is C⁎-embedded in X, then it is P-embedded in X.

For a space X, let e(X)=ω⋅sup{|D|:D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions: (1) For a product space X=∏λ∈ΛXλ, when is e(X)=|Λ|⋅sup{e(Xλ):λ∈Λ}? (2) For a Σ-product Σ of spaces Xλ,λ∈Λ, when is e(Σ)=sup{e(Xλ):λ∈Λ}? We show that the equalities in these questions hold if each Xλ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.

© 2019 Elsevier B.V. We construct a normal countably tight T1 space X with t(Xδ)>2ω. This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss [5]. We also show that if the continuum is not so large, then the tightness of Gδ-modifications of countably tight spaces can be arbitrary large up to the least ω1-strongly compact cardinal.

© 2018 Elsevier B.V. Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of P κ λ and showed that the existence of skinny stationary subsets of P κ λ is related to large cardinal properties of ideals over P κ λ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

© 2018 We show that if CH holds and either (i) there exists an ω1-Kurepa tree, or (ii) □(ω2) holds, then there are regular T1 Lindelöf spaces X0 and X1 with points Gδ such that the extent of X0×X1 is strictly greater than 2ω.

© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim We introduce a new combinatorial principle on singular cardinals. This principle allows us to take a kind of a diagonal intersection of more than λ many measure one sets of certain normal ideals over ℘(λ). Under the principle, we give various characterizations of the saturation property of normal ideals over ℘(λ). We also consider Chang's type transfer properties under the principle, and, when λ is Jónsson, we prove that every normal ideal over ℘(λ) with {x ⊆ λ : |x| = λ} measure one cannot have strong properties.

© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. We study the subtlety of a cardinal κ and of kΛ. We show that, under a certain large cardinal assumption, it is consistent that kΛ is subtle for some Λ > κ but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of PkΛ can be characterized by a certain partition relation on kΛ. We also study the property of faintness of κ, and the subtlety of kΛ with the strong inclusion.

© 2015, Springer-Verlag Berlin Heidelberg. Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ≥  3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing (Formula presented.), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.

© 2016 The Mathematical Society of Japan. Starting from a model with a weakly compact cardinal, we construct a model in which the weak stationary reflection principle for uω2 holds but the Fodor-type reflection principle for uω2 fails. So the stationary reflection principle for uω2 fails in this model. We also construct a model in which the semi-stationary reflection principle holds but the Fodor-type reflection principle for uω2 fails.

© 2015 The Mathematical Society of Japan. Let k be a regular uncountable cardinal, and λ be a cardinal greater than k. Our main result asserts that if (λ<k)<(λ<k) = λ<k, then (pk,λ(NInk,λ<k))+ → + (NS[λ]<k k;λ )+; NSk;λs+)3 and (pk,λ(NAInk;λ<k))+ → (NSk;λs+)3, where NSk;λs (respectively, NS[λ]<k k;λ) denotes the smallest seminormal (respectively, strongly normal) ideal on Pk(λ), NInk,λ<k (respectively, NAInk;λ<k) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of Pk(λ<k), and pk,λ : Pk(λ<k) → Pk(λ) is defined by pk,λ(x) = x ∩ λ.

We prove that for a compact Hausdorff space X, if λc(X)<w(X) for every infinite cardinal λ<w(X) and λc(X)<cf(w(X)) for every infinite cardinal λ<cf(w(X)), then Tikhonov cube [0,1]w(X) is a continuous image of X, in particular the cardinality of X is just 2w(X). As an application of this result, we consider elementary submodel spaces and improve Tall's result in [17]. © 2014 Elsevier B.V.

For an uncountable cardinal κ, let (†)κ be the assertion that every ω1-stationary preserving poset of size ≤κ is semiproper. We prove that (†)ω2 is a strong principle which implies a strong form of Chang's conjecture. We also show that (†)2ω1 implies that NS ω1 is presaturated. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In this paper, we study some connections between characters of countably tight spaces of size ω1 and inaccessible cardinals. A countable tight space is indestructible if every σ-closed forcing notion preserves countable tightness of the space. We show that, assuming the existence of an inaccessible cardinal, the following statements are consistent:(1)Every indestructibly countably tight space of size ω1 has character ≤ω1.(2)2ω1>ω2 and there is no countably tight space of size ω1 and character ω2. For the converse, we show that, if ω2 is not inaccessible in the constructible universe L, then there is an indestructibly countably tight space of size ω1 and character ω2. © 2013 Elsevier B.V.

© 2014 University of Houston. e improve results of Shelah, Tall, and Scheepers concerning the cardinality of Lindelöf spaces with small pseudocharacter. We establish the consistency of an analog of Borel's Conjecture for subspaces of [0; 1]@1 .

We introduce the notion of skinniness for subsets ofP ë and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2ë-saturation of NSë | X, where NSë denotes the non-stationary ideal over Pë, implies the existence of a skinny stationary subset of X. We also show that if ë is a singular cardinal, then there is no skinnier stationary subset of Pë. Furthermore, if ë is a strong limit singular cardinal, there is no skinny stationary subset of Pë. Combining these results, we show that if ë is a strong limit singular cardinal, then NSë | X can satisfy neither precipitousness nor 2ë-saturation for every stationary X Pë. We also indicate that ë(Eë <), where Eë < def = { < ë cf() < }, is equivalent to the existence of a skinnier (or skinniest) stationary subset of Pë under some cardinal arithmetical hypotheses.© 2013, Association for Symbolic Logic.

We study combinatorial large cardinal properties on P κ λ, such as ineffability, almost ineffability, subtlety, and the Shelah property. We show that, even when λ > κ, the almost ineffability of P κ λ does not yield the ineffability of κ. We also show that the Shelah property and the partition property of P κ λ do not yield the subtlety of κ. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We investigate the partition property of Pκλ. Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that Pκλ carries a (λ <κ, 2)-distributive normal ideal without the partition property, then λ is Π n1-indescribable for all n < ω but not Π 12 -indescribable. (2) If cf(λ) ≥ κ, then every ineffable subset of Pκλ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over Pκλ has the partition property. © 2012 Springer-Verlag.

Let A be a non-empty set. A set S subset of p(A) is said to be stationary in p(A) if for every f: [A](&lt;omega) -&gt; A there exists x is an element of S such that x not equal A and f"[x](&lt;omega) subset of x. In this paper we prove the following: For an uncountable cardinal 1 and a stationary set S in p(lambda), if there is a regular uncountable cardinal kappa &lt;= lambda such that {x is an element of S : x boolean AND kappa is an element of kappa} is stationary, then S can be split into kappa disjoint stationary subsets.

We study various partition properties on P(kappa)(lambda). Our main result asserts that lambda(&lt;lambda&lt;kappa) = lambda(&lt;kappa), then (p(NSh(kappa,lambda&lt;kappa)))(+) --&gt; (NSS(kappa,lambda)(+))(2), where p : P(kappa)(lambda(&lt;kappa)) --&gt; P(kappa)(lambda) is defined by p(x) = x boolean AND lambda. (C) 2011 Elsevier BM. All rights reserved.

This article is to give a brief survey of roles of large cardinals (those ordinals which are in certain sense very big) in set theory of our time. In particular, close relationship of the structure of the continuum to large cardinals is emphasized. We also mention the inner model method which is a comparable approach to large cardinal axioms, so that we could make clear the reason why large cardinals are so important.

We introduce a new reflection principle which we call "Fodor-type Reflection Principle" (FRP) This principle follows from but is strictly weaker than Fleissner&apos;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 &lt;= aleph(2)
We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelof if all of its subspaces of cardinality &lt;= aleph(1) are (Theorem 4 3) It follows that. under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality &lt;= aleph(1) 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) (C) 2009 Elsevier B V All rights reserved

For a regular uncountable cardinal kappa and a cardinal lambda with cf(lambda) &lt; kappa &lt; lambda, we investigate the consistency strength of the existence of a stationary set in P-kappa lambda which cannot be split into lambda(+) many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in P-kappa lambda such that every stationary subset of S can be split into lambda(+) many pairwise disjoint stationary subsets.

We study ineffability, the Shelah property, and indescribability of P-kappa lambda when cf(lambda) &lt; kappa. We prove that if lambda is a strong limit cardinal with cf(lambda) &lt; kappa then the ineffable ideal, the Shelah ideal, and the completely ineffable ideal over P-kappa(lambda) are the same, and that it can be precipitous. Furthermore we show that Pi(1)(1)-indescribability of P-kappa(lambda) is much stronger than ineffability if 2(lambda) = lambda(&lt;kappa).

Starting with lambda-supercompact cardinal K, where lambda is a regular cardinal greater than or equal to K, we produce a model with a stationary subset S of P-k lambda such that NSk lambda vertical bar S, the ideal generated by the non-stationary ideal NSk lambda over P-k lambda together with P-k lambda\S, is; lambda(+)-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH).
We also show that in our model we can make NSk lambda vertical bar S(k, lambda) lambda(+)-saturated, where S(k, lambda) is the set of all x epsilon P-k lambda such that ot(x), the order type of x, is a regular cardinal and x is stationary in sup(x). Furthermore we construct a model where NSk lambda vertical bar S(k, lambda) is k(+)-saturated but GCH fails. We show that if S\S(k, lambda) is stationary in P-k lambda, then S can be split into lambda many disjoint stationary subsets. (c) 2007 Elsevier B.V. All rights reserved.