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Register a new userOpenly factorizable spaces and compact extensions of topological semigroups
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We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-v{C}ech compactification $beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:Sto Y$ to a second countable space $Y$ can be written as the composition $f=gcirc p$ of an open map $p:Xto Z$ onto a second countable space $Z$ and a map $g:Zto Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.
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We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q).
We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.
We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $Stimes S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an old Wallaces result saying that each simple compact topological semigroup is a topological paragroup.
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A special class of spacial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called cartesian and studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma emph{I}, are called emph{I}-cartesian and characterized. The characterization reveals a hidden structure of such spaces. Several other characterizations are obtained and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.
Let us call a (para)topological group emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo-logical groups by means of continuous real-valued functions. We show that a (para)topo-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $fcolon Gto mathbb{R}$, one can find a continuous homomorphism $varphi$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $gcolon Hto mathbb{R}$ such that $f=gcircvarphi$. This characterization is applied for the study of completions of simply $sm$-factorizable topological groups. We prove that the equalities $mu{G}=varrho_omega{G}=upsilon{G}$ hold for each Hausdorff simply $sm$-factorizable topological group $G$. This result gives a positive answer to a question posed by Arhangelskii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $upsilon{G}$, and the Dieudonne completion, $mu{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $upsilon{G}$ is also simply $sm$-factorizable. Some results in [emph{Completions of paratopological groups, Monatsh. Math. textbf{183} (2017), 699--721}] are improved or generalized.
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