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.
In this paper, the regular ω -balanced paratopological groups with a q-point are investigated. We mainly show that: (1) a paratopological group G is a regular ω-balanced q-space with S m ( G ) ≤ ω iff for each open neighborhood O of the identity e in G there is a closed countably compact invariant subgroup H ⊆ O of G such that the quotient space G/H is a regular and quasi-metrizable paratopological group while the canonical quotient mapping p : G → G / H is open and quasi-perfect; (2) a paratopological group G is a regular ω-balanced strong q-space iff each open neighborhood O of e contains a closed sequentially compact neutral subgroup H of countable character, which is a strong q-sequence at each y ∈ H such that H ⊆ O, the quotient space G/H is a regular and quasi-metrizable paratopological group while the canonical quotient mapping p : G → G / H is open and strongly sequential-perfect; (3) a regular paratopological group G is an ω-balanced pointwise pseudocompact space with S m ( G ) ≤ ω iff for every open neighborhood O of the identity e in G, there exists a closed r-pseudocompact invariant subgroup H ⊆ O and a sequence { Wn : n ∈ ω } of open neighborhoods of H, such that every family { On : n ∈ ω } of non-empty open sets with On ⊆ Wn for each n ∈ ω has an accumulation point in H that Wn = Wn H for each n ∈ ω and the quotient space G/H is an ω-balanced pointwise pseudocompact space in which every point is a regular Gδ -set. These results extend the characterizations of q-spaces, strict q-spaces, and strong q-spaces established by [3] to the class of ω-balanced paratopological groups.
In this paper, we investigate the nearly strongly star-Menger property and investigate its relationships with other known properties in the literature. Then we characterize nearly Menger, nearly strongly star-Menger and nearly star-Menger properties in hyperspaces endowed with the hit-and-miss topology τΔ.
In this paper we present Furi Pera Schauder fixed point results for acyclic or admissible type maps between completely regular topological spaces.
The aim of this paper is to obtain fixed point results for interpolative Kannan type contraction mappings. The purpose of this paper is two fold: one relates to the fixed point results for multivalued cyclic interpolative Kannan type contractions, and second deals with proving the Perov fixed point result for interpolative Kannan type contraction mappings in the framework of vector valued metric spaces.
