Embedding paratopological groups into topological products

By Mikhail Tkachenko. 2009

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Abstract We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω -balanced (totally ω -narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that ⋂ V ∈ γ V V − 1 ⊆ U . Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω -balanced (totally ω -narrow) and the index of regularity of G is countable. As a by-product, we show that a regular totally ω -narrow paratopological group with countable index of regularity is Tychonoff.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f8324ae3ec0a4333d5698a3b56aaf94a920a07c7