We propose a notion of multi-scale stability conditions with the goal of providing a smooth compactification of the quotient of the space of projectivized Bridgeland stability conditions by the group of autoequivalence. For the case of the 3CY category associated with the $A_n$-quiver, this goal is achieved by defining a topology and complex structure that relies on a plumbing construction.

We compare this compactification to the multi-scale compactification of quadratic differentials and briefly indicate why even for the Kronecker quiver, this notion needs refinement to provide a full compactification.

We continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$ , defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$ -equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$ -divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_{\sim }$ -equivalent ultrafilters do not necessarily have the same residue modulo $m\in \mathbb {N}$ . Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we introduce a strengthening of $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ and show that it also behaves well with respect to the congruence relation.

We investigate C-sets in almost zero-dimensional spaces, showing that closed $\sigma $ C-sets are C-sets. As corollaries, we prove that every rim- $\sigma $ -compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show that these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of $($ Cantor set $)\!\times \mathbb R$ is nowhere dense.

The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\ell _{p}$ spaces for all $1\leqslant p<\infty$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.

Let A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let A s be the algebra of strong*-continuous functions from X to K(H). Then A s /A is the inner corona algebra of A. We show that if X has no isolated points, then A s /A is an essential ideal of the corona algebra of A, and Prim(A s /A), the primitive ideal space of A s /A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of A s /A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of A s /A. Several of the results are obtained in the more general setting of C 0(X)-algebras.

We answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., $\operatorname{cov}\left( \text{M} \right)=\mathfrak{c})$, a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space that does not include a perfect pre-image of ${{\text{ }\!\!\omega\!\!\text{ }}_{1}}$ is hereditarily paracompact.

In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D. K. Burke to show that there is a Scott domain $P$ for which $\max (P)$ is a ${{G}_{\delta }}$-subset of $P$ and yet no measurement $\mu $ on $P$ has $\text{ker(}\mu \text{)}\,=\,\max (P)$. We also correct a mistake in the literature asserting that $[0,\,{{\omega }_{1}})$ is a space of this type. We show that if $P$ is a Scott domain and $X\,\subseteq \,\max (P)$ is a ${{G}_{\delta }}$-subset of $P$, then $X$ has a ${{G}_{\delta }}$-diagonal and is weakly developable. We show that if $X\,\subseteq \,\max (P)$ is a ${{G}_{\delta }}$-subset of $P$, where $P$ is a domain but perhaps not a Scott domain, then $X$ is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain $P$ such that $\max (P)$ is the usual space of countable ordinals and is a ${{G}_{\delta }}$-subset of $P$ in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space.

The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in Cn into PN(C) omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings.

In this paper we study the Stone-Čech bicompactification () of the bispace (X, P, Q). We show that the ring of all continuous real-valued functions on () may be identified with the uniform closure of a suitable subring of C(). Using this result, we give a characterization of the Wallman-Sanin compactifications of the pairwise Tychonoff bitopological spaces.

In [7,3.1 ] the authors show that if a space X is realcompact and locally compact, then X* is a P′-space. In this paper we show that the hypothesis of realcompactness can be weakened. We also look at other conditions on X that are sufficient to guarantee that X* is a P′-space.

A uniform space X is said to be uniformly locally connected if given any entourage U there exists an entourage V ⊂ U such that V[x] is connected for each x ∈ X. It is said to have property S if given any entourage U, X can be written as a finite union of connected sets each of which is U-small.

Based on these two uniform connection properties, another proof is given of the following well known result in the theory of locally connected spaces: The Stone-Čech compactification βX is locally connected if and only if X is locally connected and pseudocompact.

Compactifications are constructed for convergence ordered spaces and topological ordered spaces with extension properties that resemble those of the Stone-Čech compactification.

A completely regular HausdorfT space is extra countably compact if every infinite subset of βX has an accumulation point in X. It is a theorem of Comfort and Waiveris that if X either an F-space or realcompact (topologically complete), then there is a set {Pξ:ξ<2C} of extra countably compact (countably compact) subspaces of αX such that Pξ ∩ Pξ = X, for ξ<ξ'<2C. Comfort and Waiveris conjecture that in all three cases, the spaces may be chosen pairwise non-homeomorphic. We prove this conjecture, using D- limits and weak P-points. We also give a partial solution to another question asked by Comfort and Waiveris.

In [3], Reed establishes a bijection between the (equivalence classes of) principal T1-extensions of a topological space X and the compatible, cluster-generated, Lodato nearnesses on X. We extend Reed's result to the T0 case by obtaining a one-to-one correspondence between the principal T0-extensions of a space X and the collections of sets (called “t-grill sets”) which generate a certain class of nearnesses which we call “t-bunch generated” nearnesses. This correspondence specializes to principal T0-compactifications. Finally, we show that there is a bijection between these t-grill sets and the filter systems of Thron [5], and that the corresponding extensions are equivalent.

For a completely regular Hausdorff topological space X, let Z(X) denote the lattice of zero-sets of X. If T is a continuous map from X to Y, then there is a lattice homomorphism T” from Z(Y) to Z(X) induced by T which is defined by τ‘(A) = τ←(A). A characterization is given of those lattice homomorphisms from Z(Y) to Z(X) which are induced in the above way by a continuous function from X to Y.

We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.