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A dual pair formulation for asymmetric locally convex spaces is developed that strictly generalises the ordinary vector space setting. The concept of a polar topology carries over to the asymmetric case and some familiar results are…

In this paper we prove that if f is a planar K-quasiconformal map and 0<t<2, t' = 2t/(2K-Kt+t), then f transforms sets of finite (t')-Hausdorff measure into sets of finite t-Hausdorff measure. We also prove the following more quantitative…

We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a…

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a…

We investigate how the Hausdorff dimension and measure of a self-similar set $K\subseteq\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\mathcal{T}$ generated by the orthogonal parts of the defining maps…

We determine the exact complexity of classifying compact metric spaces up to homeomorphism. More precisely, the homeomorphism relation on compact metric spaces is Borel bi-reducible with the complete orbit equivalence relation of Polish…

Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet $X$ and output alphabet $Y$ can be naturally endowed with the quotient of the Euclidean topology by the…

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever…

In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example…

Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spaces not containing a copy of $c_0$. We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions $\mathcal{C}(K_1, E_1)$ and…

We provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $\lambda$ of countable confinality. Somewhat unexpectedly, the resulting theory is…

Let $X$ be a zero-dimensional metric space and $X'$ its derived set. We prove the following assertions: (1) the space $C_k(X,2)$ is an Ascoli space iff $C_k(X,2)$ is $k_\mathbb{R}$-space iff either $X$ is locally compact or $X$ is not…

An open set U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does not define N. It follows that (R,+,x,U) defines sets in every level of the projective…

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology $\tau_{ac}$. We show that, under mild conditions on a…

We show that for each natural $n>1$ it is consistent that there is a compact Hausdorff space $K_{2n}$ such that in $C(K_{2n})$ there is no uncountable (semi)biorthogonal sequence $(f_\xi,\mu_\xi)_{\xi\in \omega_1}$ where $\mu_\xi$'s are…

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…

We show that if a separable Rosenthal compactum $K$ is an $n$-to-one preimage of a metric compactum, but it is not an $n-1$-to-one preimage, then $K$ contains a closed subset homeomorphic to either the $n-$Split interval $S_n(I)$ or the…

We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no…

For a separable locally compact but not compact metrizable space $X$, let $\alpha X = X \cup \{x_\infty\}$ be the one-point compactification with the point at infinity $x_\infty$. We denote by $EM(X)$ the space consisting of admissible…

In this paper we prove that if $\phi:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq…