The $\tau$-precompact Hausdorff Group Reflection of Topological Groups

Hei, Wei; Xiao, Zhiqiang

doi:10.36045/bbms/1523412056

Cited by 4 publications

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“…Then f has a continuous homomorphism extension pf : pG → pH such that the following diagram commutes: Let G and H be two discrete abelian groups and let f : G → H be a one-to-one homomorphism. Then each homomorphism g : G → T has an extensionĝ : The following result was proved in [6].…”

Section: The Contravariant Lattice-valued Functormentioning

confidence: 96%

Lattice of compactifications of a topological group

Xiao

2018

CGASA

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We show that the lattice of compactifications of a topological group G is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification bG of G. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained.

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Section: The Contravariant Lattice-valued Functormentioning

confidence: 96%

Lattice of compactifications of a topological group

Xiao

2018

CGASA

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“…Let us recall that the precompact Hausdorff reflection of a given topological group G is a pair (H, ϕ G ), where H is a precompact Hausdorff topological group and ϕ G : G → H is a continuous onto homomorphism, such that for every continuous homomorphism g : G → K to a Hausdorff precompact topological group K, there exists a continuous homomorphism h : H → K satisfying g = h • ϕ G . Every topological group G has a precompact Hausdorff reflection and this reflection is unique up to topological isomorphism [13]. The homomorphism ϕ G is referred to as universal for G. Proof.…”

Section: Proofmentioning

confidence: 99%

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

Tkachenko¹

2021

Preprint

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We study factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let $D=\prod_{i\in I}D_i$ be a product of paratopological groups, $S$ be a dense subgroup of $D$, and $\chi$ a continuous character of $S$. Then one can find a finite set $E\subset I$ and continuous characters $\chi_i$ of $D_i$, for $i\in E$, such that $\chi=\big(\prod_{i\in E} \chi_i\circ p_i\big)\hs1\res\hs1 S$, where $p_i\colon D\to D_i$ is the projection.

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“…Let us recall that the precompact Hausdorff reflection of a given topological group G is a pair (H, ϕ G ), where H is a precompact Hausdorff topological group and ϕ G : G → H is a continuous homomorphism, such that for every continuous homomorphism g : G → K to a Hausdorff precompact topological group K, there exists a continuous homomorphism h : H → K satisfying g = h • ϕ G . Every topological group G has a precompact Hausdorff reflection and this reflection is unique up to topological isomorphism [12]. The homomorphism ϕ G is referred to as universal for G. Lemma 4.…”

Section: Proofmentioning

confidence: 99%

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

Tkachenko¹

2021

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This study is on the factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let D=∏i∈IDi be a product of paratopological groups, S be a dense subgroup of D, and χ a continuous character of S. Then one can find a finite set E⊂I and continuous characters χi of Di, for i∈E, such that χ=∏i∈Eχi∘piS, where pi:D→Di is the projection.

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The $\tau$-precompact Hausdorff Group Reflection of Topological Groups

Cited by 4 publications

References 0 publications

Lattice of compactifications of a topological group

Lattice of compactifications of a topological group

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

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