Cardinal invariants of paratopological groups

Abstract

We show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity of G, we can find a neighborhood V of and a countable family γ of neighborhoods of in G such that W ∈γ V W −1 ⊆ U. We prove that every regular (Hausdorff) totally ω-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf number. We also prove that every Hausdorff paratopological group with countable π-character has a regular G δ -diagonal. Keywords

Products of bounded subsets of paratopological groups

“…Since H is a continuous homomorphic image of G, the space H is weakly Lindelöf. Therefore, combining [14,Theorem 2.10] and [23,Proposition 3.5], we see that the Hausdorff number of the group H is countable. Then Theorem 3.7 implies that every bounded subset of H is strongly bounded.…”

“…We claim that γ satisfies condition ( * ) of Definition 2.3. Since the group G is regular and totally ω-narrow, it embeds as a subgroup into a product of regular second countable paratopological groups [14,Corollary 2.4]. Equivalently, there exists a family L of continuous homomorphisms of G onto regular second countable paratopological groups which generates the original topology of G. Therefore, we can additionally assume that for every n ∈ ω, the set U n has the form U n = p −1 n ( U n ), where p n ∈ L and U n is an open subset of the regular second countable paratopological group p n (G).…”

Remainders in compactifications of semitopological and paratopological groups

“…Lemma 2.1 is easy, so we omit its proof. Lemma 2.2 was proved in [23], which plays an important role in the proof of Theorem 2.3.…”

“…Therefore, according to Lemma 2.8 G has countable π-character. From [23,Theorem 2.25] it follows that G has a regular G δ -diagonal. 2…”

“…1 Lemma 2.2. ( [23]) Every ω-narrow semitopological group of countable π-character has countable π-base.…”