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Register a new userNonexistence of Countable Extremally Disconnected Groups with Many Open Subgroups
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Olga Sipacheva
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It is proved that the existence of a countable extremally disconnected Boolean topological group containing a family of open subgroups whose intersection has empty interior implies the existence of a rapid ultrafilter.
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It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasovs question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.
Given an arbitrary measurable cardinal $kappa$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $kappa$ is constructed.
A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology implies that of a nondiscrete extremally disconnected group of cardinality at most $2^omega$ with linear topology.
In this paper, we prove that: (1) Let $f:Grightarrow H$ be a continuous $d$-open surjective homomorphism; if $G$ is an $mathbb{R}$-factorizabile paratopological group, then so is $H$. Peng and Zhangs result cite[Theorem 1.7]{PZ} is improved. (2) Let $G$ be a regular $mathbb{R}$-factorizable paratopological group; then every subgroup $H$ of $G$ is $mathbb{R}$-factorizable if and only if $H$ is $z$-embedded in $G$. This result gives out a positive answer to an question of M.~Sanchis and M.~Tkachenko cite[Problem 5.3]{ST}.
A Hausdorff topological space $X$ is called $textit{superconnected}$ (resp. $textit{coregular}$) if for any nonempty open sets $U_1,dots U_nsubseteq X$, the intersection of their closures $bar U_1capdotscapbar U_n$ is not empty (resp. the complement $Xsetminus (bar U_1capdotscapbar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $mathbb Qmathsf P^infty$ of the topological vector space $mathbb Q^{<omega}={(x_n)_{ninomega}in mathbb Q^{omega}:|{ninomega:x_n e 0}|<omega}$ over the field of rationals $mathbb Q$. The space $mathbb Qmathsf P^infty$ is the quotient space of $mathbb Q^{<omega}setminus{0}^omega$ by the equivalence relation $xsim y$ iff $mathbb Q{cdot}x=mathbb Q{cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $mathbb Qmathsf P^infty$, and a topological space $X$ is homeomorphic to $mathbb Qmathsf P^infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{ninomega}$ such that (i) $X_0=X$, $bigcap_{ninomega}X_n=emptyset$, (ii) for every $ninomega$ and a nonempty open set $Usubseteq X_n$ the closure $bar U$ contains some set $X_m$, and (iii) for every $ninomega$ the complement $Xsetminus X_n$ is a regular topological space. Using this topological characterization of $mathbb Qmathsf P^infty$ we find topological copies of the space $mathbb Qmathsf P^infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.
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