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Displaying similar documents to “More on ordinals in topological groups”

Ordinals in topological groups

Raushan Z. Buzyakova (2007)

Fundamenta Mathematicae

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We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the cellularity and index of narrowness in topological groups and their G δ -modifications. We show, in particular, that the inequalities in ( ( H ) τ ) 2 τ · in ( H ) and c ( ( H ) τ ) 2 2 τ · in ( H ) hold for every topological group H and every cardinal τ ω , where ( H ) τ denotes the underlying group H endowed with the G τ -modification of the original topology of H and in ( H ) is the index of narrowness of the group H . Also, we find some bounds for the complexity of continuous real-valued functions f on an arbitrary ω -narrow group...

Arhangel'skiĭ sheaf amalgamations in topological groups

Boaz Tsaban, Lyubomyr Zdomskyy (2016)

Fundamenta Mathematicae

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We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property α 1 . 5 is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space C p ( X ) of continuous real-valued functions on X with the...

Classification of spaces of continuous functions on ordinals

Leonid V. Genze, Sergei P. Gul'ko, Tat'ana E. Khmyleva (2018)

Commentationes Mathematicae Universitatis Carolinae

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We conclude the classification of spaces of continuous functions on ordinals carried out by Górak [Górak R., Function spaces on ordinals, Comment. Math. Univ. Carolin. 46 (2005), no. 1, 93–103]. This gives a complete topological classification of the spaces C p ( [ 0 , α ] ) of all continuous real-valued functions on compact segments of ordinals endowed with the topology of pointwise convergence. Moreover, this topological classification of the spaces C p ( [ 0 , α ] ) completely coincides with their uniform classification. ...

On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique

Alejandro Ramírez-Páramo (2005)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If X is a T 1 space such that (i) L ( X ) t ( X ) κ , (ii) ψ ( X ) 2 κ , and (iii) for all A [ X ] 2 κ , A ¯ 2 κ , then | X | 2 κ ; and (b) (Fedeli [2]) If X is a T 2 -space then | X | 2 aql ( X ) t ( X ) ψ c ( X ) .

Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, Lewis C. Robertson

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For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly...

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, U α : α , such that U α U β whenever β ≤ α for all α , β . The class of all metrizable topological groups is a proper subclass of the class T G of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group G T G is metrizable, and hence G is strictly angelic. We deduce from...