Papers

We study the notion of a super caliber of a topological space, which is closely related to the classical notion of caliber and has appeared in the literature under several different names. We collect and unify several known results and establish new results concerning the collections of super calibers of topological spaces and their hyperspaces. In particular, we investigate the relationship between the super calibers of a space $X$ and those of hyperspaces $\mathcal{H}(X)$ lying between $\mathrm{CL}(X)$ and $\mathcal{F}(X)$. For infinite metrizable spaces, we characterize several cases in which $\mathsf{SC}(X)$ and $\mathsf{SC}(\mathrm{CL}(X))$ differ and establish an independence result over ZFC; see Theorem~5.12.

We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over \(\I=[0,1]\). Using the Lowen fuzzy Sierpiński object \(\Sier\), which identifies \(\tau_X\) with \(C(X,\Sier)\), we explicitly determine the largest splitting topology on this mapping set. Its open weights \(\Phi:\tau_X\to\I\) are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of \(\Sier\). This yields an intrinsic characterization: \(X\) is exponentiable if and only if every \(\mu\in\tau_X\) satisfies \[ \mu=\bigvee_{\lambda\triangleleft\Phi} (\const{\Phi(\mu)}\wedge\lambda), \qquad \lambda\triangleleft\Phi \Longleftrightarrow \const{\Phi(\nu)}\wedge\lambda\leq\nu \quad(\nu\in\tau_X). \] When this condition holds, \(Y^X\) has underlying set \(C(X,Y)\), with topology generated by \([\Phi,v](f)=\Phi(v\circ f)\). We also obtain a dual closed-set formulation and three applications. Exponentiability implies that \(\tau_X\) is a continuous lattice, although the converse fails. Moreover, a classical space \(X\) is exponentiable exactly when its induced fuzzy space \(\omega X\) is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and \(N\)-compact Hausdorff spaces are exponentiable.

The Asplund property plays an important role in Banach space theory due to its connections with differentiability properties of continuous convex functions, optimization problems, and the weak topology of Banach spaces. Motivated by the variety of nonequivalent definitions proposed in the literature for locally convex spaces, in this survey we provide a unified framework for studying the Asplund property beyond the Banach space setting. The main object of our study is the locally convex space of continuous functions $C_k(X)$ endowed with the compact-open topology, where $X$ is an arbitrary Tychonoff space. We completely characterize the Asplund property for $C_k(X)$ in terms of topological properties of the underlying space $X$. As an essential step, we revisit the proof of several classical results, including the Namioka–Phelps theorem. Our approach is independent of differentiability techniques and relies solely on topological methods. The exposition is self-contained, and all major results are provided with complete proofs, making the paper accessible to both specialists and newcomers.

For infinite cardinals $\lambda \leq \kappa$, we calculate the calibers of topological spaces $X$ of cardinality $\kappa$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < \lambda$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.

Let $A$ be an abelian group and $E=\Endo_{\Z}(A)$. We give a module-theoretic description of the singular $E$-submodules asked for in Fuchs' Problem~1.2. For a unital ring $R$ and a left $R$-module $M$, put $W=I_R({}_RR)\oplus I_R(M)$ and $J=\Jaco(\Endo_R(W))$, and let $u$ be the image of $1_R$. The classical essential-kernel criterion yields \[ Z_R(M)=M\cap Ju. \] For $R=E$ and $M=A$, all singular submodules are therefore the $E$-submodules of $A\cap Ju$. Writing $T=t(A)$ and $B=A/T$, we prove a torsion-transfer formula, identify the torsion part as $\bigoplus_p pT_p$, and describe simultaneous prime lifting by a canonical obstruction. The resulting extension gives an $\Extt/\Homm$ parametrization of all singular submodules. We obtain explicit formulas for torsion groups and for $\Z(p^\infty)\oplus B$ with $B$ torsion-free; in the latter case the fully invariant subgroup lattice of $B$ occurs as an interval. The general description retains the induced endomorphism action and extension data, rather than giving a classification by classical group invariants.

We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.

Zavarzina asked whether every bijective non-contractive self-map of a precompact quasi-metric space is an isometry. We answer the question in the negative, in a stronger form: on a hereditarily precompact quasi-metric space (a tolled extension of the irrational rotation over Z), the shift is a bijective non-contractive map that is not an isometry. The construction generalizes: for every metric space Y, every bijective isometry $\sigma$ of Y, and every point whose $\sigma$-orbit is aperiodic and totally bounded, the tolled Z-extension is such a counterexample; instances include the irrational rotation and the p-adic odometer. The conjugate space is not precompact, so the two-sided hypothesis of Zavarzina's positive result is sharp, and the space is not expand-contract plastic. The core results, including the general scheme and the circle instance, are machine-checked in Lean 4 over mathlib, sorry-free (no unproved steps), under a pinned toolchain. A structural section proves a monochromy theorem: for every strict pair on a hereditarily precompact space, one endpoint's backward self-returns have self-defect at least a quarter of the pair's defect, uniformly on a chain-closed displacement set. Boundary results delimit what remains open: one-sided compactness together with separation.

A Stieltjes clock allows effective time to advance continuously, remain unchanged over intervals, or jump. We study whether a compact-valued set-valued map evolving relative to a Stieltjes clock admits a single-valued selection that passes through a prescribed graph point while retaining the regularity and variation of the multifunction. For g-Hölder exponents $\alpha \geq 1$, we prove that regularity can be preserved without increasing the g-Hölder seminorm, with no monotonicity assumption on g. If g is additionally nondecreasing, one prescribed-point selection preserves both this regularity and the Hausdorff variation on every subinterval. The same selection consequently preserves all finite Riesz p-variations associated with nondecreasing external clocks. For $\alpha>1$, the structure becomes jump-driven. For left-continuous nondecreasing Stieltjes clocks, continuous clock evolution cannot generate variation: all variation is carried by jumps. We obtain exact jump decompositions for both the set-valued map and its selection, together with explicit atomic formulas for Riesz p-variation. Examples show that the principal regularity, variation, and jump bounds are attained. For compact-convex Euclidean-valued maps, we also examine the case $0<\alpha<1$. In this setting, the Hölder exponent can still be preserved, but preservation of the same constant through a prescribed point holds in one dimension and can fail in higher dimensions. Finally, without a quantitative Hölder bound, we characterize exactly when every compact-valued Hausdorff g-continuous map admits a prescribed-point g-continuous selection: precisely when the clock image is zero-dimensional.

It is known that, for ordinary Hölder multifunctions with 0 < {\alpha} < 1, Hölder regularity alone does not guarantee the existence of a Hölder selection, and in general even a continuous selection may fail to exist. We ask how this situation changes when the parameter is measured through a Stieltjes clock. For 0 < {\alpha} < 1, we characterize the clocks for which every compact-valued g-Hölder multifunction admits a g-Hölder selection. The determining condition is uniform disconnectedness of the clock image Im(g). For left-continuous and nondecreasing clocks, this condition is equivalent to uniform jump dominance: every positive clock increment contains a jump carrying a fixed positive fraction of that increment. Under this condition, a selection can be chosen through any prescribed point of the graph, with explicit control of its Hölder regularity. Conversely, when the condition fails, there exists a compact-valued g-Hölder multifunction with no g-Hölder selection. The results show that the selection property is governed not simply by the presence of jumps, but by how their sizes are distributed across scales.

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

In this survey, we review certain types of special \(T_{1}\) spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary \(T_{0}\) spaces. Furthermore, the \(T_{1}\) separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.

We give a short proof that $\kappa$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable $\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.

Let $(X,\tau)$ be metrizable and let $F=\{a_1,\ldots,a_k\}\subseteq X$, where $2\le k<\infty$. We represent all metrizable topologies $\sigma\subseteq\tau$ agreeing with $\tau$ on $X\setminus F$ by compatible systems of continuous gauges $s_i:X\to[0,1]$ with $s_i^{-1}(0)=\{a_i\}$. The condition $\inf_X\max\{s_i,s_j\}>0$ for $i\ne j$ is equivalent to both Hausdorffness and metrizability of the prescribed gauge topology. A normalized product map into the standard simplex gives an explicit metric; its triangle inequality follows from a simplex slack inequality. This metric is complete whenever the auxiliary bounded compatible metric is complete. For two compatible systems, their coordinatewise minimum describes the intersection topology. It is compatible exactly when the two coarsenings have a common Hausdorff lower bound; in that case the intersection is metrizable and is their meet. Otherwise every common lower topology is non-Hausdorff. A closed-discrete construction produces such an obstructed pair for every noncompact metrizable space and every finite exceptional set with at least two points. Consequently, for these exceptional sets, the family is downward directed, or is a lattice, if and only if $(X,\tau)$ is compact, in which case it consists only of $\tau$.

We use metric functionals to build a weak topology associated with $d$-weak convergence on metric spaces. We also show the existence of a sequence in the space $C[0,1]$ that is unbounded in the sup-norm yet converges $d$-weakly. This points out an error in an earlier work.

Assuming Jensen's diamond principle, we construct an infinite compact zero-dimensional space $K$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. The space $K$ is separable and crowded, has weight $\aleph_1$ and cardinality $2^{\aleph_1}$, and is an Efimov space. We construct $K$ as an inverse limit of compact metrisable spaces indexed by the countable ordinals. At each nontrivial successor step, the projection has two-point fibres over a chosen closed set and singleton fibres elsewhere; this changes the weak-star limit of a selected sequence of finitely supported measures. We also prove that, for compact $X$, the existence of an infinite-dimensional separable quotient of $C_p(X)$ is equivalent to the existence of an infinite-dimensional metrisable quotient.

A compactification of the discrete space $\omega$ of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of $\omega$. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of $\omega$, and the Arkhangel'skiĭ–Bereznitskiĭ–Schachermayer space studied in $C_p$-theory, which is in a sense the maximal symmetric compactification of $\omega$. We investigate symmetric compactifications of $\omega$ and their relations to the Separable Quotient Problem for spaces $C_p(X)$ and to the existence of Josefson–Nissenzweig sequences of finitely supported Borel measures on spaces $X$, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces $C_p(K)$, $K$ compact, to the case when $K$ is a totally asymmetric compactification of $\omega$. Our results shed some new light on the Grothendieck property of Banach spaces $C(K)$.

The topology of a bounded three-dimensional domain can strongly affect electromagnetic fields. In a topologically complex domain, a curl-free field may not have a globally single-valued scalar potential and a divergence-free field may also fail to have a global vector potential. These topological effects lead naturally to harmonic vector fields in the Helmholtz decomposition. This paper gives a geometric and constructive study of such fields using vector analysis, circulation integrals, cutting surfaces, scalar Laplace problems, Stokes' theorem, and Green's identity. The first Betti number is interpreted through independent handle-type circulations, whereas the second Betti number counts enclosed voids. Direct proofs are given for the dimensions of the Neumann and Dirichlet harmonic-field spaces. The analysis is also extended to anisotropic lossless media. The results provide a simple topological interpretation of harmonic fields and physical DC modes in bounded electromagnetic resonators.

In this paper, we study the upper box dimensions, the packing dimensions and the quasisymmetric minimalities of some homogeneous Moran sets. We obtain the upper box dimension formula and the packing dimension formula for three classes of homogeneous Moran sets under some conditions. And two classes of them with packing dimension 1 are shown to be quasisymmetrically packing-minimal.

Our main result is a construction of a class H of pathwise connected, locally connected, compact Hausdorff spaces such that (i) if X,Y are distinct spaces in H then a continuous, injective mapping from X to Y does not exist; (ii) H contains 2^k spaces of weight k and size k for every cardinal number k not smaller than the cardinality of the continuum; (iii) if k,l are infinite cardinals and l is not greater than k then H contains 2^k spaces of weight k and size k^l; (iv) the metrizable continua in H are subspaces of the plane.

In this paper, we study the notion of $\mathcal{I}$$^{st}$-convergence of sequences of real numbers which is an extension work of $\mathcal{I}$$^*$-convergence on real sequences. We also define the convergence field of $\mathcal{I}$$^{st}$-convergence and $\mathcal{I}$$^{st}$-limit points and we discuss some properties related to these notions.

Let $\Gamma$ be a countable discrete $C^*$-simple group. Let $\mu$ be a $\Gamma$-invariant Borel probability measure on $\text{Sub}(\Gamma)$, i.e., an IRS. We show that $\mu$-almost every subgroup $H$ is $C^*$-simple.

In this paper, we further investigate the role of the graded Grothendieck group $K_0^{\gr}$ and its positive cone (the talented monoid) as an effective tool for distinguishing structural types of algebras associated to higher-rank $k$-graphs. We study the prime spectrum (the space of all prime $\Gamma$-order ideals equipped with a Zariski-like topology) of a general commutative $\Gamma$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $\Gamma$-order ideal is finitely generated. As a result we are able to show that the prime spectrum of the talented monoid of a row-finite $k$-graph without sources and with a finite set of vertices is spectral. For any row-finite $k$-graph $\Lambda$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian–Pask algebra $\KP(\Lambda)$ is homeomorphic to both the space of all prime $\mathbb{Z}^k$-order ideals and the space of all prime $\mathbb{Z}^k$-filters of the talented monoid $T_\Lambda$. Another main result of this paper provides a complete topological description of regular $\Gamma$-order ideals of a refinement $\Gamma$-monoid: a $\Gamma$-order ideal $J$ is regular if and only if the corresponding closed (resp., open) set $V(J)$ (resp., $D(J)$) is regular closed (resp., regular open) in the prime spectrum. As an application of these results, we establish a lattice isomorphism between the lattice of all regular $\mathbb{Z}^k$-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian–Pask algebra via a spectrum-theoretic approach.

Every compact subset of a homogeneous generalized ordered (GO) space has character at most $\omega_1$ and cardinality at most $2^{\omega_1}$; if such a subset has uncountable character, then the character of the whole space equals $\omega_1$ and its $\pi$-character is countable. We construct a homogeneous $\sigma$-compact linearly ordered space (LOTS) $\mathbf{H}$ containing a compact subset $\mathbf{S}$ of cardinality $2^{\omega_1}$ whose character is $\omega_1$ at every point and whose weight and Souslin number are both $2^{\omega_1}$; thus both bounds obtained are sharp. We prove that a semitopological group that is a GO space is hereditarily paracompact; if, in addition, it is not a $P$-space, then it is submetrizable, has countable character, and its compact subsets are metrizable. Every linearly ordered semitopological group (and, more generally, every GO semitopological group) is either metrizable or is a $P$-space; the same holds for topological groups. We also show that in an order-homogeneous LOTS every compact subset is first countable.

Let $R$ be a commutative ring with nonzero identity and let $R^\#$ denote the set of its nonzero nonunits. We extend the divisor topology $D(R)$, previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology $GD(R)$ on $EC(R^\#)$. Its basic open sets are \[ B_a=\{[b]\in EC(R^\#): b\mid a^n \text{ for some }n\geq 1\}. \] The relation \[ [b]\in B_a \quad\Longleftrightarrow\quad \sqrt{aR}\subseteq\sqrt{bR} \] shows that $GD(R)$ records radical divisibility among principal ideals. We prove that $GD(R)$ is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the $T_0$ and discrete properties and of the equality $GD(R)=D(R)$. We also determine the isolated points of $GD(R)$. Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain $R$, $GD(R)$ is compact if and only if $R$ is a $G$-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of $GD(R)$ with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.

V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that $P_\tau(X)$ is $C$-embedded in $P_\sigma(X)$; (ii) Is it true that $P_R(X)$ is $C$-embedded in $P_R(\beta X)$ if and only if $X$ is pseudocompact, where $P_\sigma$, $P_\tau$, and $P_R$ are the functors of probability $\sigma$-additive on the Baire $\sigma$-algebra, $\tau$-additive, and Radon measures on the space $X$? The answers to these questions are negative. However, if instead of probability measures we consider the corresponding alternating measures $M_\sigma$, $M_\tau$, and $M_R$, the situation changes. It is proved that (i) $M_\tau(X)$ is $C$-embedded in $M_\sigma(X)$; (ii) $M_R(X)$ is $C$-embedded in $M_R(\beta X)$ if and only if $X$ is pseudocompact. The question of $C$-embedding of measure spaces is an extension of the question of coincidence of measure spaces, which is a development of the classical concepts of universally measurable and universal measure zero sets. A general theorem is obtained, which leads to the mentioned results.

For a finite set $S$ with $\lvert S\rvert = N$, the number of families $\mathcal{B} \subseteq \mathcal{P}(S)$ that are topological bases is $\#(N) = \sum_{\mathcal{T} \in \operatorname{Top}(S)} 2^{\lvert\mathcal{T}\rvert - \lvert\mathcal{M}_{\mathcal{T}}\rvert}$, where $\mathcal{M}_{\mathcal{T}}$ is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating $\mathcal{T}$ are exactly the sets with $\mathcal{M}_{\mathcal{T}} \subseteq \mathcal{B} \subseteq \mathcal{T}$. The small-$N$ table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.

In this paper, we study the computational complexity of finding fixed points for set-valued contractions. We first formulate a computational problem for Nadler's fixed-point theorem: Projected-Nadler, and prove that it is $\mathsf{CLS}$-complete by showing its equivalence to Continuous-LocalOpt. We then establish a stronger converse for Nadler's fixed point theorem that can be applied as a tool to analyze the convergence rate of set-valued basic iteration procedure. Finally, we reduce large-margin triplet stationarity problem to Projected-Nadler. Together with its $\mathsf{CLS}$-hardness introduced in [arXiv:2509.16898], this yields $\mathsf{CLS}$-completeness of large-margin triplet stationarity.

This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $\kappa$-barrelled and $\omega$-barrelled groups. Our first main result establishes that every $\omega$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $\sigma$-compact, $\omega$-barrelled group, then its dual group $\widehat{G}_\kappa$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $\omega$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.

Let $(X,\tau)$ be metrizable and let $a\in X$. We give a constructive account of metrizable topologies $\sigma\subseteq\tau$ that agree with $\tau$ on $X\setminus\{a\}$. Applying Hausdorff's classical metric collapse construction, for every noncompact $(X,\tau)$ and every compatible metric $d$ we obtain a strict coarsening with a metric $p\le d$ that agrees with $d$ on a common neighborhood of each point other than $a$. A prescribed countably infinite closed discrete set $\{x_n:n\in\N\}\subseteq X\setminus\{a\}$ can be made to satisfy $p(a,x_n)\le\lambda_n$ for any positive null sequence $(\lambda_n)$. The resulting metric is greatest among the metrics dominated by $d$ that satisfy these bounds, and is complete whenever $d$ is complete. We exhibit its realization as a classical metric quotient. We also represent all localized metrizable coarsenings by continuous scalar gauges using a standard cone metric. Inclusion is expressed by the cofinal comparison of sublevel sets familiar from extension-trace theory, while pointwise maximum and minimum realize finite joins and meets. A closed-discrete criterion detects strictness. Standard preservation results for Borel structure, complete metrizability, and Polishness, together with function-space and local-field examples, complete the account.

We show that maximal abelian C*-subalgebras (masas) of the Calkin algebra (the algebra of all bounded operators on the separable Hilbert space modulo compact operators) may consistently have their densities strictly less than continuum and we describe many isomorphism types of such masas. Specifically, we prove that after adding any number of Cohen reals to a model of CH the algebra $C(K_{\mathcal A})$ of all complex-valued continuous functions on the Stone space $K_{\mathcal A}$ of a Boolean algebra $\mathcal A$ of cardinality $\omega_1$ is $*$-isomorphic to a masa of the Calkin algebra if and only if $\mathcal A$ does not admit a countably generated ultrafilter. Moreover, for every such Boolean algebra we obtain $\omega_2$ pairwise unitarily non-equivalent such masas, none of which has a commutative lift. We also show in ZFC that if a C*-algebra of the form $C(K)$ for any compact Hausdorff $K$ is $*$-isomorphic to a masa of the Calkin algebra, then no point of $K$ may have character smaller than $\mathfrak p$. Therefore, consistently, there may not be any masa of the Calkin algebra of density less than continuum.