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1.
Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup Hω12 that is an HFD with the following property
(P)
the projection of H onto every partial product I2 for Iω[ω1] is onto.
Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ2, there is an HFD topological group in ω12 which has property (P).  相似文献   

2.
On box products     
We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <ω1?□α<ω1Xα where each Xα is a compact metric space).  相似文献   

3.
We introduce a general method of resolving first countable, compact spaces that allows accurate estimate of inductive dimensions. We apply this method to construct, inter alia, for each ordinal number α>1 of cardinality ?c, a rigid, first countable, non-metrizable continuum Sα with . Sα is the increment in some compactification of [0,1) and admits a fully closed, ring-like map onto a metric continuum. Moreover, every subcontinuum of Sα is separable. Additionally, Sα can be constructed so as to be: (1) a hereditarily indecomposable Anderson-Choquet continuum with covering dimension a given natural number n, provided α>n, (2) a hereditarily decomposable and chainable weak Cook continuum, (3) a hereditarily decomposable and chainable Cook continuum, provided α is countable, (4) a hereditarily indecomposable Cook continuum with covering dimension one, or (5) a Cook continuum with covering dimension two, provided α>2.We also produce a chainable and hereditarily decomposable space Sω(c+) with , , trind0Sω(c+) and trInd0Sω(c+) all equal to ω(c+), the first ordinal of cardinality c+.  相似文献   

4.
For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(ω2) and p(ωω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X)=p(ω2) or p(X)=p(ωω). We show that the following statements are each relatively consistent with ZFC: (a) p=p(ωω)<p(ω2) and (b) p<p(ωω)=p(ω2)  相似文献   

5.
A space X is said to be selectively separable (=M-separable) if for each sequence {Dn:nω} of dense subsets of X, there are finite sets FnDn (nω) such that ?{Fn:nω} is dense in X. On selective separability and its variations, we show the following: (1) Selective separability, R-separability and GN-separability are preserved under finite unions; (2) Assuming CH (the continuum hypothesis), there is a countable regular maximal R-separable space X such that X2 is not selectively separable; (3) c{0,1} has a selectively separable, countable and dense subset S such that the group generated by S is not selectively separable. These answer some questions posed in Bella et al. (2008) [7].  相似文献   

6.
It is well known that for dynamical systems generated by continuous maps of a graph, the centre of the dynamical system is a subset of the set of ω-limit points.In this paper we provide an example of a continuous self-map f1 of a dendrite such that ω(f1) is a proper subset of C(f1).The second example is a continuous self-map f2 of a dendrite having a strictly increasing sequence of ω-limit sets which is not contained in any maximal one. Again, this is impossible for continuous maps on graphs.  相似文献   

7.
We show that CH implies that P(ω), when equipped with the Vietoris topology, has a subspace which is an L-space and a subspace which is an S-space. This is an immediate consequence of the following purely combinatorial result: CH implies the existence of an ω1-sequence 〈xα: α < ω1〉 in P(ω) such that (1) if α<β<ω1, then Xβ?1Xα; (2) if I ?ω1 is unaccountable, then there are distinct α, β ∈ I with Xβ ?Xα.  相似文献   

8.
Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supnHAn is uncountable for every Hω[N] then ?nGAn is uncountable for some Gω[N]. This fact, by our definition, means that the σ-ideal [X]?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation EX2 of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.  相似文献   

9.
A family {Mα|α?A} is a shrinking of a cover {Oα|α?A} of a topological space if {Mα|α?A} also covers and Mα?Oα for all α?A.?++ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking.? implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.  相似文献   

10.
We study the space of linear orders on a given set X, denoted by Op(X), endowed with the topology of pointwise convergence. We show, in particular, that if |X|=ω1 or |X|=ω0 then Op(X) is homeomorphic to ω12 and ω02, respectively.  相似文献   

11.
We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ω1]<ω. We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.  相似文献   

12.
We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness ofXωand that of the hyperspaceX2? We do that first in the context of Mrówka-Isbell spaces Ψ(A) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space Ψω(A) is pseudocompact for every MAD family A. We show that
(1)
(p=c) 2Ψ(A) is pseudocompact for every MAD family A.
(2)
(h<c) There is a MAD family A such that 2Ψ(A) is not pseudocompact.
We also construct a ZFC example of a space X such that Xω is pseudocompact, yet X2 is not.  相似文献   

13.
For an ordinal α, α2 denotes the collection of all nonempty closed sets of α with the Vietoris topology and K(α) denotes the collection of all nonempty compact sets of α with the subspace topology of α2. It is well known that α2 is normal iff cfα=1. In this paper, we will prove that for every nonzero-ordinal α:
(1)
α2 is countably paracompact iff cfαω.
(2)
K(α) is countably paracompact.
(3)
K(α) is normal iff, if cfα is uncountable, then cfα=α.
In (3), we use elementary submodel techniques.  相似文献   

14.
A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed 02. Consequently, a first countable linearly Lindelöf space is Lindelöf if ω>02. They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ω. We answer this question in the negative by constructing a counterexample from MA+ω<02.A modification of Alster?s Michael space that is first countable is presented.  相似文献   

15.
In this paper the structure of hereditarily strong Σ-spaces (hsΣ-spaces, for short) is dealt with. The main result asserts that an hsΣ-space is the disjoint union of two σ subspaces one of which is an Fσ, the other a Gδ subset. Examples are given that in many ways, this decomposition cannot be improved. Then we investigate the question when an hsΣ-space is a σ-space. It is shown that a GO-space (or a first countable compactum) is metrizable iff it is an hsΣ-space, thereby proving a conjecture of J. van Wouwe. σ-spaces are characterized as being identical with perfect hsΣ-spaces. The question whether a Lindelöf, first countable hsΣ-space is a σ-space is shown to be independent of set theory. A characterization of hsΣ-spaces with no compact subsets of cardinality >2ω is given.  相似文献   

16.
Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin?(O,Ω) (the latter means that for every sequence 〈unnω of open covers of T there exists a sequence 〈vnnω such that vn∈[un]<ω and for every F∈[X]<ω there exists nω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.  相似文献   

17.
A space X is κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ2-resolvable but not ω1-resolvable space of dispersion character κ.  相似文献   

18.
It is shown that ω × Yω does not have remote points if Y is a compact space with cellularity larger than ω1. It is also shown that it is consistent that ω × Yω does not have remote points if Y is compact with uncountable cellularity. As an application we construct a compact space with weight ω2 · c which can be covered by nowhere dense P-sets and a compact space with weight c for which it is independent that it can be covered by nowhere dense P-sets.  相似文献   

19.
We show that the transfinite inductive dimensions modulo PP-trind and P-trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A(α) or the absolutely multiplicative Borel class M(α), 0?α<ω1.  相似文献   

20.
We define the multiple zeta function of the free Abelian group Zd as
ζZd(s1,…,sd)=∑|Zd:H|<α1(H)s1?αd(H)sd,  相似文献   

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