Countably compact group topologies on non-torsion abelian groups from selective ultrafilters / Topologias enumeravelmente compactas em grupos abelianos de não torção via ultrafiltros seletivos
AUTOR(ES)
Ana Carolina Boero
FONTE
IBICT - Instituto Brasileiro de Informação em Ciência e Tecnologia
DATA DE PUBLICAÇÃO
11/03/2011
RESUMO
Assuming the existence of $\\mathfrak c$ pairwise incomparable selective ultrafilters (according to the Rudin-Keisler ordering) we prove that the free abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology that contains a non-trivial convergent sequence. Under the same hypothesis, we show that an abelian almost torsion-free topological group $(G, +, \\tau)$ with $|G| = |\\tau| = \\mathfrak c$ admits a group topology independent of $\\tau$ and we algebraically characterize the non-torsion abelian groups of cardinality $\\mathfrak c$ which admit a countably compact group topology (without non-trivial convergent sequences). We also prove that the free abelian group of cardinality $\\mathfrak c$ admits a group topology that makes its square countably compact and we construct a Wallace\ s semigroup whose square is countably compact. Finally, assuming the existence of $2^{\\mathfrak c}$ selective ultrafilters, we ensure that if a non-torsion abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology, then it admits $2^{\\mathfrak c}$ (pairwise non-homeomorphic) countably compact group topologies.
ASSUNTO(S)
compacidade enumerável countable compactness grupos topológicos selective ultrafilters topological groups ultrafiltros seletivos
