Non-trivial Convergent Sequences Research Articles

Convergent sequences in iterated ultrapowers as p-compact groups

We prove, in ZFC, that if G is an infinite countable Abelian group, then there is an ultrafilter p∈ω⁎ such that (Ultp(G),τBohr‾) has non-trivial convergent sequences, consequently (Ultpω1(G),τBohr‾) has non-trivial convergent sequences, extending Theorem 3.9 from [13]. In addition, we prove that the Remark 3.8 from [13] is false; so, the proof of the Corollary 3.11 is false too.

The Josefson–Nissenzweig theorem and filters on $$\omega $$

AbstractFor a free filter F on $$\omega $$ ω , endow the space $$N_F=\omega \cup \{p_F\}$$ N F = ω ∪ { p F } , where $$p_F\not \in \omega $$ p F ∉ ω , with the topology in which every element of $$\omega $$ ω is isolated whereas all open neighborhoods of $$p_F$$ p F are of the form $$A\cup \{p_F\}$$ A ∪ { p F } for $$A\in F$$ A ∈ F . Spaces of the form $$N_F$$ N F constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space $$N_F$$ N F carries a sequence $$\langle \mu _n:n\in \omega \rangle $$ ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that $$\mu _n(f)\rightarrow 0$$ μ n ( f ) → 0 for every bounded continuous real-valued function f on $$N_F$$ N F if and only if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z , that is, the dual ideal $$F^*$$ F ∗ is Katětov below the asymptotic density ideal $${\mathcal {Z}}$$ Z . Consequently, we get that if $$F^*\le _K{\mathcal {Z}}$$ F ∗ ≤ K Z , then: (1) if X is a Tychonoff space and $$N_F$$ N F is homeomorphic to a subspace of X, then the space $$C_p^*(X)$$ C p ∗ ( X ) of bounded continuous real-valued functions on X contains a complemented copy of the space $$c_0$$ c 0 endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$N_F$$ N F is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.

Some pseudocompact-like properties in certain topological groups

We construct in ZFC a countably compact group without non-trivial convergent sequences of size 2c, answering a question of Bellini, Rodrigues and Tomita [4]. We also construct in ZFC a selectively pseudocompact group which is not countably pracompact, showing that these two properties are not equivalent in the class of topological groups. Using the same technique, we construct a group which has all powers selectively pseudocompact but is not countably pracompact, assuming the existence of a selective ultrafilter.

Small cardinals and small Efimov spaces

We introduce and analyze a new cardinal characteristic of the continuum, the splitting number of the reals, denoted s(R). This number is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial convergent sequence, or else a copy of βN.

MORE ON FRÉCHET–URYSOHN IDEALS

AbstractWe study the Rudin–Keisler pre-order on Fréchet–Urysohn ideals on $\omega $ . We solve three open questions posed by S. García-Ferreira and J. E. Rivera-Gómez in the articles [5] and [6] by establishing the following results: •For every AD family $\mathcal {A},$ there is an AD family $\mathcal {B}$ such that $\mathcal {A}^{\perp } <_{{\textsf {RK}}}\mathcal {B}^{\perp }.$ •If $\mathcal {A}$ is a nowhere MAD family of size $\mathfrak {c}$ then there is a nowhere MAD family $\mathcal {B}$ such that $\mathcal {I}\left (\mathcal {A}\right ) $ and $\mathcal {I}\left ( \mathcal {B}\right ) $ are Rudin–Keisler incomparable.•There is a family $\left \{ \mathcal {B}_{\alpha }\mid \alpha \in \mathfrak {c}\right \} $ of nowhere MAD families such that if $\alpha \neq \beta $ , then $\mathcal {I}\left ( \mathcal {B}_{\alpha }\right ) $ and $\mathcal {I}\left ( \mathcal {B}_{\beta }\right ) $ are Rudin–Keisler incomparable.Here $\mathcal {I}(\mathcal {A})$ denotes the ideal generated by an AD family $\mathcal {A}$ .In the context of hyperspaces with the Vietoris topology, for a Fréchet–Urysohn-filter $\mathcal {F}$ we let $\mathcal {S}_{c}\left ( \mathcal {\xi }\left ( \mathcal {F}\right ) \right ) $ be the hyperspace of nontrivial convergent sequences of the space consisting of $\omega $ as discrete subset and only one accumulation point $\mathcal {F}$ whose neighborhoods are the elements of $\mathcal {F}$ together with the singleton $\{\mathcal {F}\}$ . For a FU-filter $\mathcal {F}$ we show that the following are equivalent: • $\mathcal {F}$ is a FUF-filter.• $\mathcal {S}_{c}\left ( \mathcal {\xi }\left ( \mathcal {F} \right ) \right ) $ is Baire.

Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Assuming the existence of c incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality c that admit a countably compact group topology. We show that for each κ∈[c,2c] each of these groups has a countably compact group topology of weight κ without non-trivial convergent sequences and another that has convergent sequences.Assuming the existence of 2c selective ultrafilters, there are at least 2c non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most 2c is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality c that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.

Forcing a classification of non-torsion Abelian groups of size at most [formula omitted] with non-trivial convergent sequences

We force a classification of all the Abelian groups of cardinality at most 2c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most 2c, by showing that if a non-torsion Abelian group of size at most 2c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

On countably compact group topologies without non-trivial convergent sequences on [formula omitted] for arbitrarily large κ and a selective ultrafilter

Castro Pereira and Tomita showed that it is consistent that there exist arbitrarily large countably compact groups without non-trivial convergent sequences of finite order. In this paper, we obtain large p-compact groups without non-trivial convergent on the direct sum of κ copies of Q where κ=κω is infinite. In particular this gives the first arbitrarily large countably compact groups without non-trivial convergent sequences that are torsion-free.

The Baire property on the hyperspace of nontrivial convergent sequences

In this paper, we shall consider the hyperspace of all nontrivial convergent sequences Sc(X) of a Fréchet-Urysohn nondiscrete space X, which is equipped with the Vietoris topology. We study the spaces X for which Sc(X) is Baire: this kind of spaces have a dense subset of isolated points (see [12]). We characterize the spaces X for which Sc(X) is Baire. This characterization uses a topological game inspired by the Banach-Mazur game. As a consequence, we obtain that if X is completely metrizable and has a dense subset of isolated points, then Sc(X) is Baire. Our last main result shows that Sc(X) is pseudocompact iff X is homeomorphic to the one-point compactification of a discrete space of uncountable size. This last assertion provides a characterization of the one-point compactification of a discrete space of uncountable size.

Countably compact groups without non-trivial convergent sequences

Countably compact groups without non-trivial convergent sequences

On feebly compact paratopological groups

We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each 2-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topologically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a T0 sequentially compact group, a T1 2-pseudocompact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free Abelian countably compact topological group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group G is feebly compact provided it has a feebly compact normal subgroup H such that a quotient group G/H is feebly compact. For our research we also study some general constructions of paratopological groups. We extend the well-known construction of Raĭkov completion of a T0 topological group to the class of paratopological groups. We investigate cone topologies of paratopological groups which provide a general tool for constructing pathological examples, especially examples of compact-like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup S of a group G and compact-like properties of two basic semigroup topologies generated by S on the group G.

Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

Let K be a compact Hausdorff space and let C(K) be the space of all scalar-valued, continuous functions on K. We show that C(K) is an $$\ell _1(K)$$ -Grothendieck space but not a Grothendieck space exactly when the spaces $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are not linearly isomorphic, where $${\mathbb {N}}^{\#}$$ is the one-point compactificiation of the discrete space of natural numbers. (That is, if C(K) contains a complemented copy of $$c_0$$ , then C(K) fails to be $$\ell _1(K)$$ -Grothendieck if and only if the topologies of pointwise convergence in $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are linearly isomorphic.) Moreover, for infinite compact spaces K and L, there exists a compact space G that has a non-trivial convergent sequence and such that $$C_{p}(K\times L)$$ and $$C_{p}(G)$$ are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided.

Contractibility of the hyperspace of sequences, harmonic fan

Given a Hausdorff space X, let Sc(X) be the hyperspace of nontrivial convergent sequences. It is an open problem to determine if the contractibility of X implies the contractibility of Sc(X). In this paper we prove that if X is the harmonic fan, then Sc(X) is contractible. We also prove that the contractibility of X implies that Sc(X) is pathwise connected. This answers questions by Camargo, Maya and Pellicer-Covarrubias.

On convergent sequences in dual groups

We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} groups $G$, we characterize the existence of a nontrivial convergent sequence in $G_p^\wedge$ by the following property of $G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present an example of a dense subgroup $G$ of the compact metrizable group $\mathbb{Z}(2)^\omega$ such that $G$ is of the first category in itself, has measure zero, but the dual group $G_p^\wedge$ does not contain infinite compact subsets. This complements Theorem 1.6 in [J.E.~Hart and K.~Kunen, Limits in function spaces and compact groups, \textit{Topol. Appl.} \textbf{151} (2005), 157--168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.

Path connectedness, local path connectedness and contractibility of $\mathcal{S}_c(X)$

The hyperspace of all nontrivial convergent sequences in a Hausdorff space $X$ is denoted by $\mathcal {S}_c(X)$. This hyperspace is endowed with the Vietoris topology. In connection with a question and a problem by García-Ferreira, Ortiz-Castillo an

Cardinal invariants and convergence properties of locally minimal groups

Let G be a locally essential subgroup of a locally compact abelian group K. Then:(i)t(G)=χ(G)=χ(K), where t(G) and χ(G) are the tightness and the character of G, respectively;(ii)if G is radial, then K must be metrizable;(iii)if G is non-discrete, then G contains a super-sequence S converging to 0 such that |S|=χ(G)=χ(K); in particular, G has non-trivial convergent sequences.Items (i)–(iii) hold when G is a dense locally minimal subgroup of K. It follows that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007.For every uncountable cardinal κ, we construct a Fréchet-Urysohn minimal nilpotent group G of nilpotency class 2 and character κ such that the connected component of G is an open normal ω-bounded subgroup of G (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.

Countably compact group topologies on non-torsion Abelian groups of size continuum with non-trivial convergent sequences

Under p=c, we answer Question 24 of [9] for cardinality c, by showing that if a non-torsion Abelian group of size continuum admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences

We show that if κ≤ω and there exists a group topology without non-trivial convergent sequences on an Abelian group H such that Hn is countably compact for each n<κ then there exists a topological group G such that Gn is countably compact for each n<κ and Gκ is not countably compact. If in addition H is torsion, then the result above holds for κ=ω1. Combining with other results in the literature, we show that:a) Assuming c incomparable selective ultrafilters, for each n∈ω, there exists a group topology on the free Abelian group G such that Gn is countably compact and Gn+1 is not countably compact. (It was already know for ω).b) If κ∈ω∪{ω}∪{ω1}, there exists in ZFC a topological group G such that Gγ is countably compact for each cardinal γ<κ and Gκ is not countably compact.

Cardinal functions of the hyperspace of convergent sequences

Abstract The symbol 𝓢 c (X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π-weight, the π-character, the pseudocharacter and the Lindelöf number of 𝓢 c (X) with the corresponding cardinal function of X. We also answer a question posed by the authors in a previous paper.

Weakly Radon–Nikodým Boolean algebras and independent sequences

A compact space is said to be weakly Radon-Nikodým (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated nonWRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodriguez and R. Haydon concerning measures and the existence of nontrivial convergent sequences on WRN compacta, obtaining partial results on some natural subclasses.