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.
We study forcing properties of the Boolean algebras P(ω)/I, where I is a Borel ideal on ω. We show (Theorem 2.12) that (under a large cardinal hypothesis) P(ω)/I does not add reals if and only if it has a dense σ-closed subset. For analytic P-ideals I we show (Theorem 3.3) that either P(ω)/I is ωω-bounding or it is not proper. We also investigate the existence of completely separable I-MAD families.
We compute the exact descriptive class of the set of all compact arc-connected subsets of R2, which turns out to be strictly higher than the classical Sigma 11 and II11 classes of analytic and coanalytic sets, but strictly lower than the class II12which is the exact descriptive class of the set of all compact arc-connected subsets of R3.
We study symmetries in equivariant versions of Khovanov homology, which include (i) the construction of an involution sigma b for the U(2)-equivariant theory, (ii) an integral lifting nu b of the Shumakovitch operation nu, and (iii) splitting of the U specialIntscript-and U specialIntscript & times; U(1)-equivariant theories generalizing earlier work over F 2 . Finally, we relate these structures to the Rasmussen s-invariant over an arbitrary field F.
We investigate the problem of finding the minimum number of pieces necessary for dividing a three-dimensional sphere or a ball and reassembling it to form n congruent copies of the original object, generalising a known result by Raphael Robinson.
Thom polynomials provide universal formulas for the fundamental class of singularity loci in terms of characteristic classes. Ohmoto extended this notion to SSM-Thom polynomials, which refine this description by capturing the richer Segre-Schwartz-MacPherson (SSM) class of singularity loci. While previous methods for computing SSM-Thom polynomials relied on intricate geometric arguments, we introduce a more efficient approach that depends solely on the symmetries of singularities. Our method is inspired by connections to Geometric Representation Theory, particularly the interpolation properties of Maulik-Okounkov stable envelopes. By formulating SSM analogs of these axioms within a degree-bounded framework, we obtain new computational tools for SSM-Thom polynomials. We also present explicit examples of SSM-Thom polynomials, and illustrate their applications in enumerative geometry and singularity theory.
A regular separable first-countable countably compact space is called a Nyikos space. In this paper, we give a partial solution to an old problem of Nyikos by showing that each locally compact Nyikos inverse topological semigroup is compact. Also, we show that a topological semigroup S that contains a dense inverse subsemigroup is a topological inverse semigroup, provided (i) S is compact, or (ii) S is countably compact and sequential. The latter result solves a problem of Banakh and Pastukhova and provides the automatic continuity of inversion in certain compact-like inverse semigroups.
We show that there exists a class of symbolic subshifts which realizes all Choquet simplices as simplices of invariant measures and the conjugacy relation on that class is hyperfinite.
From many supercompact cardinals, we show that it is consistent for the tree property to hold at many small successors of singular cardinals, each with a different cofinality. In particular, we construct a model in which the tree property holds at aleph(omega+omega +1) and at aleph(omega n+1) for all 0 < n < omega. We show that this can be done for the strong tree property as well, and extend the technique to large uncountable sequences of desired cofinalities.
Kauffman virtual knots are knots in thickened surfaces F×R considered up to isotopy, stabilizations and destabilizations, and diffeomorphisms of F×R induced by orientation preserving diffeomorphisms of F. Similarly, virtual Legendrian knots, introduced by Cahn and Levi, are Legendrian knots in ST∗F with the natural contact structure. Virtual Legendrian knots are considered up to isotopy, stabilization and destabilization of the surface away from the front projection of the Legendrian knot, as well as up to contact isomorphisms of ST∗F induced by orientation preserving diffeomorphisms of F. We show that there is a projection operation proj from the set of virtual isotopy classes of Legendrian knots to the set of isotopy classes of Legendrian knots in ST∗S2. This projection is obtained by substituting some of the classical crossings of the front diagram with virtual crossings. It restricts to the identity map on the set of virtual isotopy classes of classical Legendrian knots. In particular, proj extends invariants of Legendrian knots to invariants of virtual Legendrian knots. Using proj, we show that the virtual crossing number of every classical Legendrian knot equals its crossing number. We also prove that the virtual canonical genus of a Legendrian knot is equal to the canonical genus. The construction of proj is inspired by the work of Manturov.
We solve a problem of Aliaga and Perneck & aacute; about Lipschitz free spaces (denoted by F()): Does Jevery Borel measure mu on a complete metric space M such that d(m, 0) d|mu|(m) < infinity induce a weak(& lowast; )continuous functional L is an element of F(M) by the mapping L mu (f) = integral f d mu? In particular, we obtain a characterization of the Borel measures mu such that L mu 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 mu such that L mu 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.
Let epsilon denote the a-ideal generated by closed null sets on the reals. We show that the uniformity and the covering of E can be added to Cicho & nacute;'s maximum with distinct values. More specifically, it is consistent that N-1 < add(N) < cov(N) < b < non(epsilon) < non(M) < cov(M) < cov(epsilon) < d < non(N) < cof(N) < 2(0)(N) holds.
We prove that the product of any family of pseudofinite structures is pseudofinite using the fundamental work on products of first-order structures due to Feferman and Vaught (1959), exploiting the underlying combinatorics.
We continue the study of the Erdős-Dushnik-Miller theorem (A graph with an uncountable set of vertices has either an infinite independent set or an uncountable clique) in set theory without the axiom of choice. We show that there are three inequivalent versions of this theorem and we give some results about the positions of these versions in the deductive hierarchy of weak choice principles.
We generalise the concept of topo-isomorphic extensions and define finite topomorphic extensions as topological dynamical systems whose factor map to the maximal equicontinuous factor is measure-theoretically at most m-to-one for some m is an element of N. We further define multivariate versions of mean equicontinuity, complementing the notion of multivariate mean sensitivity introduced by Li, Ye and Yu, and then show that any m-to-one topomorphic extension is mean (m + 1)-equicontinuous. This falls in line with the well-known result, due to Downarowicz and Glasner, that strictly ergodic systems are isomorphic extensions if and only if they are mean equicontinuous. While in the multivariate case we can only conjecture that the converse direction also holds, the result provides an indication that multivariate equicontinuity properties are strongly related to finite extension structures. For minimal systems, an Auslander-Yorke type dichotomy between multivariate mean equicontinuity and multivariate mean sensitivity is shown as well.
Generalizing a result of T & ouml;rnquist and Weiss, we study the connection between the existence of Sigma 12 Sierpirski's coverings of II8n, and a cardinal invariant of the upper semi-lattice of constructibility degrees known as breadth.
