Related papers: Uniform Eberlein compactifications of metrizable s…

We discuss some notions of compactness and convergence relative to a specified family F of subsets of some topological space X. The two most interesting particular cases of our construction appear to be the following ones. (1) The case in…

We describe the order type of range sets of compact ultrametrics and show that an ultrametrizable infinite topological space $(X, \tau)$ is compact iff the range sets are order isomorphic for any two ultrametrics compatible with the…

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$…

We view ultrametric spaces as two-sorted structures consisting of a set of points and of a linearly ordered set of distances. We call the appropriate notion of embeddings distance-carrying (dc for short). Those are obtained by combining…

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the…

We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally…

P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a "point at infinity") if and only if it is non-compact. He also asked for characterizations of spaces which have one-point…

Let c=2^aleph0 denote the cardinality of the continuum and let a,b,k be infinite cardinal numbers with a<b\leq 2^a. We show that there exist precisely 2^b T0-spaces of size a and weight b up to homeomorphism. Among these non-homeomorphic…

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is…

We show that if a topological or paratopological group $G$ contains a stationary subset of some regular uncountable cardinal, then $G$ contains a subspace which is not collectionwise normal. This statement implies that if a monotonically…

We prove that every topological action of a countable group on a metrizable space can be realized as a bi-Lipschitz action with respect to some compatible metric. This extends a result due to U. Hamenst\"{a}dt regarding finitely generated…

We study the presence of abelian discrete symmetries in globally consistent orientifold compactifications based on rational conformal field theory. We extend previous work [1] by allowing the discrete symmetries to be a linear combination…

The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.

We bound the rate of uniform convergence in compact sets for both entropic potentials and their gradients towards the Brenier potential and its gradient, respectively. Both results hold in the quadratic Euclidean setting for absolutely…

The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set $C$. It is well known that there are compact Hausdorff spaces of cardinality equal to that of $C$ that are not continuous…

We show that the first homology group of a locally connected compact metric space is either uncountable or is finitely generated. This is related to Shelah's well-known result which shows that the fundamental group of such a space satisfies…

The aim of the paper is to characterize (pre)compactness in the spaces of Lipschitz/H\"older continuous mappings acting from a compact metric space to a normed space. To this end some extensions and generalizations of already existing…

We introduce the open degree of a compact space, and we show that for every natural number n, the separable Rosenthal compact spaces of degree n have a finite basis.

Dimensional types of metric scattered spaces are investigated. Revised proofs of Mazurkiewicz-Sierpi\'nski and Knaster-Urbanik theorems are presented. Embeddable properties of countable metric spaces are generalized onto uncountable metric…

We say that a Tychonoff space $X$ is a $\kappa$-space if it is homeomorphic to a closed subspace of $C_p(Y)$ for some locally compact space $Y$. The class of $\kappa$-spaces is strictly between the class of Dieudonn\'{e} complete spaces and…