Zero-sets and closed sets with countable closure index
- 1Universidad Autónoma Metropolitana
Abstract
The closure index of a nonempty subset A of a semitopological group G, denoted by ci(A,G), is defined as the least cardinality of a
family γ of open neighborhoods of the identity in G such that clGA can be obtained as the intersection of the family of sets AU−1 (U−1A)
for U ∈γ. The topological and algebraic features of semitopological groups are reflected in the new cardinal function. As defined, a closed subset of a topological group G with countable closure index is a Gδ-set. The class of Gδ-sets in a topological space X is in a tight relationship with the class of zero-sets. It is well known that for a closed subset of a normal space, being a zero-set and a Gδ-set are equivalent. We show as first result that this equivalence is also true for a closed subset A of a Hausdorff topological group with countable closure index. In contrast, we show that there exists a Hausdorff paratopological Abelian group with the Lindel¨of property for which there is a closed subset that fails to be a zero-set. We also show that for every paratopological group G, ci(G) is not greater than the hereditary Lindel¨of degree of G. These results together show that there is no immediate generalization of the first result to the class of paratopological groups. Finally, we present some conditions for topological products of semitopological groups or paratopological groups to have countable closure index of open sets.
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BibTeX
@article{martinez2026zerosets,
title = {{Zero-sets and closed sets with countable closure index}},
author = {Jonás Martínez},
journal = {Applied General Topology},
year = {2026},
month = oct,
doi = {10.4995/agt.26357},
url = {https://doi.org/10.4995/agt.26357},
note = {CC BY-NC-ND},
}
RIS
TY - JOUR
TI - Zero-sets and closed sets with countable closure index
AU - Jonás Martínez
PY - 2026
DA - 2026/10/26
JO - Applied General Topology
DO - 10.4995/agt.26357
UR - https://doi.org/10.4995/agt.26357
AB - The closure index of a nonempty subset A of a semitopological group G, denoted by ci(A,G), is defined as the least cardinality of a family γ of open neighborhoods of the identity in G such that clGA can be obtained as the intersection of the family of sets AU−1 (U−1A) for U ∈γ. The topological and algebraic features of semitopological groups are reflected in the new cardinal function. As defined, a closed subset of a topological group G with countable closure index is a Gδ-set. The class of Gδ-sets in a topological space X is in a tight relationship with the class of zero-sets. It is well known that for a closed subset of a normal space, being a zero-set and a Gδ-set are equivalent. We show as first result that this equivalence is also true for a closed subset A of a Hausdorff topological group with countable closure index. In contrast, we show that there exists a Hausdorff paratopological Abelian group with the Lindel¨of property for which there is a closed subset that fails to be a zero-set. We also show that for every paratopological group G, ci(G) is not greater than the hereditary Lindel¨of degree of G. These results together show that there is no immediate generalization of the first result to the class of paratopological groups. Finally, we present some conditions for topological products of semitopological groups or paratopological groups to have countable closure index of open sets.
SN - 1576-9402
SN - 1989-4147
ER -