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Register a new userExtremally Disconnected Groups of Measurable Cardinality
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Authors
Olga Sipacheva
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Given an arbitrary measurable cardinal $kappa$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $kappa$ is constructed.
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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.
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.
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.
A.V.Arkhangelskii asked in 1981 if the variety $mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety $mathfrak V$ is a proper subclass of the class of all topological groups. A topological group $G$ is called $g$-sequential if for any topological group $H$ any sequentially continuous homomorphism $Gto H$ is continuous. We introduce the concept of a $g$-sequential cardinal and prove that a locally compact group is $g$-sequential if and only if its local weight is not a $g$-sequential cardinal. The product of a family of non-trivial $g$-sequential topological groups is $g$-sequential if and only if the cardinal of this family is not $g$-sequential. Suppose $G$ is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then $G$ is $g$-sequential if and only if its weight is not a $g$-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.
Two separated realcompact measurable spaces $(X,mathcal{A})$ and $(Y,mathcal{B})$ are shown to be isomorphic if and only if the rings $mathcal{M}(X,mathcal{A})$ and $mathcal{M}(Y,mathcal{B})$ of all real valued measurable functions over these two spaces are isomorphic. It is furthermore shown that any such ring $mathcal{M}(X,mathcal{A})$, even without the realcompactness hypothesis on $X$, can be embedded monomorphically in a ring of the form $C(K)$, where $K$ is a zero dimensional Hausdorff topological space. It is also shown that given a measure $mu$ on $(X,mathcal{A})$, the $m_mu$-topology on $mathcal{M}(X,mathcal{A})$ is 1st countable if and only if it is connected and this happens when and only when $mathcal{M}(X,mathcal{A})$ becomes identical to the subring $L^infty(mu)$ of all $mu$-essentially bounded measurable functions on $(X,mathcal{A})$. Additionally, we investigate the ideal structures in subrings of $mathcal{M}(X,mathcal{A})$ that consist of functions vanishing at all but finitely many points and functions vanishing at infinity respectively. In particular, we show that the former subring equals the intersection of all free ideals in $mathcal{M}(X,mathcal{A})$ when $(X,mathcal{A})$ is separated and $mathcal{A}$ is infinite. Assuming $(X,mathcal{A})$ is locally finite, we also determine a pair of necessary and sufficient conditions for the later subring to be an ideal of $mathcal{M}(X,mathcal{A})$.
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