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We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort.
3Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters
Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters
Convergent sequences in iterated ultrapowers as p-compact groups
Convergent sequences in iterated ultrapowers as p-compact groups
10Square of countably compact groups without non-trivial convergent sequences
Square of countably compact groups without non-trivial convergent sequences
25Chapter 18 - Efimov's problem
Chapter 18 - Efimov's problem
2S2 and the Fréchet property of free topological groups
S2 and the Fréchet property of free topological groups
3Accessible and Biaccessible Points in Contrasequential Spacesa
ABSTRACT. Two countable spaces having no nontrivial convergent sequences are constructed. One space has every point biaccessible (by a countable discrete set), and the other has every point accessible but not biaccessible (by a countable discrete set). It is shown that in a compact T2‐space if a set A is not closed, then there exists a free sequence contained in A whose closure is not contained in A. It follows that in a compact T2‐space of countable tightness, every nonisolated point is biaccessible.
Read more12Groups without convergent sequences
Groups without convergent sequences
200Approximate Diagonals and Cohomology of Certain Annihilator Banach Algebras
In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite
Read more4A Generalization of E. Siebert’s Theorem on Convergence of Convolution Semigroups and Accompanying Laws
The convergence of sequences of convolution semigroups of probabilities in terms of their generating functionals is described. The equivalence of the convergence of the accompanying laws and the corresponding generating functionals is proved in the case of an arbitrary locally compact group.
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