Convergent sequences in topological groups

Hrušák, Michael; Shibakov, Alexander

doi:10.1016/j.apal.2020.102910

Cited by 3 publications

(4 citation statements)

References 62 publications

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“…Since τ ⊆ τ 0 , every compact subspace of τ 0 is also compact in τ . Property (13) implies that every closed discrete subspace L = L γ of (B, τ 0 ) for some γ < c is also closed and discrete in τ . This in turn implies that every sequence converging in τ is also converging in τ 0 , so the sequential coreflection of (B, τ ) is (B, τ 0 ) and τ is remotely sequential.…”

Section: Nonsequential Groups and Sequential Coreflectionmentioning

confidence: 99%

See 1 more Smart Citation

Invariant ideal axiom, beyond the countable sequential groups

Hrušák,

Shibakov

2024

Fund. Math.

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Section: Nonsequential Groups and Sequential Coreflectionmentioning

confidence: 99%

“…Existence of (a rich supply of) convergent sequences in topological groups has been a subject of active research for several decades (see [1,2,7,9,16,17,18,19,21,36] and also surveys [24,25,13]). The two early questions that guided the development of this field were asked by V. Malykhin and P. Nyikos in 1978 and1980 respectively.…”

mentioning

confidence: 99%

Invariant ideal axiom, beyond the countable sequential groups

Hrušák,

Shibakov

2024

Fund. Math.

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“…Indeed, recall that every ultrafilter on the Boolean algebra induces a homomorphism from into the two-element algebra , and vice versa , and thus the Stone space of may be seen as the space , endowed with the pointwise topology originating from the Fréchet–Nikodym metric , induced by the trivial measure on . As the literature concerning convergence properties of sequences in Stone spaces of Boolean algebras, or more generally in totally disconnected topological spaces, is quite vast, cf., e.g., [5, 12, 13], we were motivated to conduct similar research in the setting of the spaces for general ’s and various topologies. Note that similar generalizations of topological spaces are quite common in analysis, e.g., one can think of Radon measures of norm on compact spaces as generalizations of points in those spaces, or, as it is done in non-commutative topology, of projections in general C*-algebras as being analogous to characteristic functions of clopen subsets of locally compact spaces.…”

Section: Introductionmentioning

confidence: 99%

On Sequences of Homomorphisms Into Measure Algebras and the Efimov Problem

Borodulin–Nadzieja¹,

Sobota²

2021

J. symb. log.

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For given Boolean algebras $\mathbb {A}$ and $\mathbb {B}$ we endow the space $\mathcal {H}(\mathbb {A},\mathbb {B})$ of all Boolean homomorphisms from $\mathbb {A}$ to $\mathbb {B}$ with various topologies and study convergence properties of sequences in $\mathcal {H}(\mathbb {A},\mathbb {B})$ . We are in particular interested in the situation when $\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb {A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\mathcal {H}(\mathbb {A},\mathbb {B})$ for a Boolean algebra $\mathbb {B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\mathbb {A}$ .

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“…e.g. [BD19], [Har07], [HS20], we were motivated to conduct similar research in the setting of the spaces H(A, M κ ) for general κ's and various topologies. Note that similar generalizations of topological spaces are quite common in analysis, e.g.…”

Section: Introductionmentioning

confidence: 99%

On sequences of homomorphisms into measure algebras and the Efimov Problem

Borodulin–Nadzieja¹,

Sobota²

2021

Preprint

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For given Boolean algebras A and B we endow the space H(A, B) of all Boolean homomorphisms from A to B with various topologies and study convergence properties of sequences in H(A, B). We are in particular interested in the situation when B is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on H(A, B) for a Boolean algebra B carrying a strictly positive measure and convergence properties of sequences of measures on A.

show abstract

Convergent sequences in topological groups

Cited by 3 publications

References 62 publications

Invariant ideal axiom, beyond the countable sequential groups

Invariant ideal axiom, beyond the countable sequential groups

On Sequences of Homomorphisms Into Measure Algebras and the Efimov Problem

On sequences of homomorphisms into measure algebras and the Efimov Problem

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