For each countable ordinal alpha, we introduce an ideal conv alpha and use it to characterize the class of all compact countable spaces which are homeomorphic to the space omega alpha & centerdot; n + 1 with the order topology. The characterization is expressed in terms of finding a convergent subsequence defined on a set not belonging to conv alpha.
Generalizing results of Dikranjan and Tkachenko (2001) and Dikranjan and Uspienskij (2023) we study the fine structure of locally minimal (locally) precompact Abelian groups (these are the locally essential subgroups G of LCA groups L, i.e., such that G non-trivially meets all "small" closed subgroup of L). More precisely we prove that if G is a dense locally minimal and sequentially closed subgroup of a LCA group L, then the connected component c(G) of G has the same weight as c(L). Moreover, when w(c(G)) is not Ulam measurable, then c(G) = c(L). We provide an extended discussion illustrating how this result fails in various ways in the non-abelian case (even for nilpotent groups of class 2). Motivated by the above result, we study further those locally minimal precompact Abelian groups G, termed critical locally minimal, such that c(G) = c(K) (where K is the compact completion of G) and G/c(G) is not locally minimal. Such a group cannot be compact, neither connected, nor totally disconnected. We provide a proper class of critical locally minimal groups with additional compactness-like properties and we study the class CClm of compact Abelian groups with a dense critical locally minimal subgroup. In particular, we completely describe the connected components of the finite-dimensional groups belonging to CClm.
Historically, logicians have found it very difficult to understand the structure of uncountable models of arithmetic. I argue that if a model M of arithmetic has an element a with |[0, a]M | = aleph 0, then it resembles a countable model. As evidence, I present two theorems. The first states that if M is such a model with its standard system contained in some Scott set S and X is an element of S, then M admits an elementary extension N with X is an element of SSy(N) subset of S. The second states that if M is such a model and L is a finite lattice such that every countable M0 has an elementary extension N0 with Lt(N0/M0) similar to= L, then M can also be extended this way. An alternative proof of Ehrenfeucht's lemma is also included.
We study ultrafilters on regular uncountable cardinals, with a primary focus on w1, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform ultrafilter over w(1) is Tukey-equivalent to [2(aleph 1)](
We solve a problem of Aliaga and Perneck & aacute; about Lipschitz free spaces (denoted by F(center dot)): Does every Borel measure & micro; on a complete metric space M such that integral d(m, 0) d|& micro;| (m) < infinity induce a weak(& lowast;) continuous functional L & micro; is an element of F(M) by the mapping L & micro;(f) = integral f d & micro;? In particular, we obtain a characterization of the Borel measures & micro; such that L & micro; is an element of F(M), which indeed implies inner-regularity for complete metric spaces. We also prove that every Borel measure on M induces an element of F(M) if and only if the weight of M is strictly less than the least real-valued measurable cardinal, and thus the existence of a metric space on which there is a measure & micro; such that L & micro; is an element of F(M)(& lowast;& lowast;) \ F(M) cannot be proven in ZFC. Finally, we partially solve a problem of Aliaga on whether every sequentially normal functional on Lip(0 )(M) is normal.
