ON -UNFAVOURABLE SPACES

To study when a paratopological group becomes a topological group, Arhangel’skii et al. [‘Topological games and topologies on groups’, Math. Maced. 8 (2010), 1–19] introduced the class of $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable spaces. We show that every $\unicode[STIX]{x1D707}$-complete (or normal) $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable semitopological group is a topological group. We prove that the product of a $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable space and a strongly Fréchet $(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$-favourable space is $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable. We also show that continuous closed irreducible mappings preserve the $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourableness in both directions.

A note on compact-like semitopological groups

We present a few results related to separation axioms and automatic continuity of operations in compact-like semitopological groups. In particular, is provided a semiregular semitopological group $G$ which is not $T_3$. We show that each weakly semiregular compact semitopological group is a topological group. On the other hand, constructed examples of quasiregular $T_1$ compact and $T_2$ sequentially compact quasitopological groups, which are not paratopological groups. Also we prove that a semitopological group $(G,\tau)$ is a topological group provided there exists a Hausdorff topology $\sigma\supset\tau$ on $G$ such that $(G,\sigma)$ is a precompact topological group and $(G,\tau)$ is weakly semiregular or $(G,\sigma)$ is a feebly compact paratopological group and $(G,\tau)$ is $T_3$.

Mapping i2 on the free paratopological groups

Let FP(X) be the free paratopological group over a topological space X. For each nonnegative integer n ? N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and in by the natural mapping from (X ? X?1 ? {e})n to FPn(X). We prove that the natural mapping i2:(X ? X?1 d ?{e})2 ? FP2(X) is a closed mapping if and only if every neighborhood U of the diagonal ?1 in Xd x X is a member of the finest quasi-uniformity on X, where X is a T1-space and Xd denotes X when equipped with the discrete topology in place of its given topology.

Free paratopological groups

Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a~neighborhood base at the identity of the group FP(X) when the space X is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group FP(X) is T_0 if X is T_0, we characterize the spaces X for which the group FP(X) is a topological group and then we give a class of spaces $X$ for which the group FP(X) has the inductive limit property.

A note on the continuity of the inverse in paratopological groups

The problem when a paratopolgical group (or semitopological group) is a topological group is interesting and important. In this paper, we continue to study this problem. It mainly shows that: (1) Let G be a paratopological group and put τ = ωHs(G); then G is a topological group if G is a Pτ-space; (2) every co-locally countably compact paratopological group G with ωHs(G) ≦ ω is a topological group; (3) every co-locally compact paratopological group is a topological group; (4) each 2-pseudocompact paratopological group G with ωHs(G) ≦ ω is a topological group. These results improve some results in [11, 13].

A note on pseudobounded paratopological groups

Let(1) There exists a metrizable, zero-dimensional and pseudobounded topological group;(2) There exists a Hausdorff ω-pseudobounded paratopological group(3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.

Cardinal invariants of paratopological groups

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 e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-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 Gσ-diagonal.