Ordinals in topological groups

Raushan Z. Buzyakova. "Ordinals in topological groups." Fundamenta Mathematicae 196.2 (2007): 127-138. <http://eudml.org/doc/286129>.

@article{RaushanZ2007,
abstract = {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 statements: (1) If τ and τ + 1 embed closedly in G then τ × (τ + 1) embeds closedly in G; (2) If τ embeds in G, G is Abelian, and the order of every non-neutral element of G is greater than $2^\{N\} - 1$ then $∏_\{i∈N\}τ$ embeds in G; (3) The previous statement holds if τ is replaced by τ + 1; (4) If G is Abelian, algebraically generated by τ + 1 ⊂ G, and the order of every element does not exceed $2^\{N\} - 1$ then $∏_\{i∈N\}(τ+1)$ is not embeddable in G.},
author = {Raushan Z. Buzyakova},
journal = {Fundamenta Mathematicae},
keywords = {topological group; space of ordinals; hereditarily normal},
language = {eng},
number = {2},
pages = {127-138},
title = {Ordinals in topological groups},
url = {http://eudml.org/doc/286129},
volume = {196},
year = {2007},
}

TY - JOUR
AU - Raushan Z. Buzyakova
TI - Ordinals in topological groups
JO - Fundamenta Mathematicae
PY - 2007
VL - 196
IS - 2
SP - 127
EP - 138
AB - 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 statements: (1) If τ and τ + 1 embed closedly in G then τ × (τ + 1) embeds closedly in G; (2) If τ embeds in G, G is Abelian, and the order of every non-neutral element of G is greater than $2^{N} - 1$ then $∏_{i∈N}τ$ embeds in G; (3) The previous statement holds if τ is replaced by τ + 1; (4) If G is Abelian, algebraically generated by τ + 1 ⊂ G, and the order of every element does not exceed $2^{N} - 1$ then $∏_{i∈N}(τ+1)$ is not embeddable in G.
LA - eng
KW - topological group; space of ordinals; hereditarily normal
UR - http://eudml.org/doc/286129
ER -