Related papers: A Core Decomposition of Compact Sets in the Plane

We refer here to the surprising construction made by Giuseppe Peano in 1890. He gave an example of a continuous function (called now the Peano curve) from the unit interval to the whole unit square. We show here the existence of a more…

This paper is the text of a lecture given at the Catania conference on Commutative Algebra and Algebraic Geometry, in honor of the 60th birthday of Silvio Greco. It analyses the correspondences between equivalence classes of objects related…

Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes…

We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space $X$ is a topological fractal if there exists $\mathcal{F}$ a finite family of self-maps on $X$ such that…

This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is a set that is convex in a designated set of convex variables when the others are held…

Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to…

A compactum $X\subset \C$ is unshielded if it coincides with the boundary of the unbounded component of $\C\sm X$. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are…

In this paper, we study the problem of computing a homotopy from a planar curve $C$ to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau's problem. Chambers and…

We show that every non-degenerate homogeneous plane continuum is homeomorphic to either the unit circle, the pseudo-arc, or the circle of pseudo-arcs. It follows that any planar homogenous compactum has the form $X \times Z$, where $X$ is a…

The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\partial K$. Let ${\cal A}(K)$ denote the space of all functions $f$ on $K$…

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the…

We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor,…

We present a quick approach to computing the $K$-theory of the category of locally compact modules over any order in a semisimple $\mathbb{Q}$-algebra. We obtain the $K$-theory by first quotienting out the compact modules and subsequently…

We construct Peano curves $\gamma : [0,\infty) \to \mathbb{R}^2$ whose "footprints" $\gamma([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent to a common continuous line field on the punctured plane $\mathbb{R}^2 \setminus…

We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is…

This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is a set function which is finitely additive on the collection of open and compact sets, inner regular on open sets, and…

Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the…

A compact space $X$ is said to be minimal if there exists a map $f:X\to X$ such that the forward orbit of any point is dense in $X$. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha, and Tywoniuk [J. Dyn.…

Let $P$ be a finite simplicial comple with underlying space (union of simplices in $P$) $|P|$. Let $Q$ be a subcomplex of $P$. Let $a \geq 0$. Then there exists $K < \infty$, \emph{depending only on $a$ and $Q$,} with the following…

A space $X$ is "sequentially $n$-connected" at $x\in X$ if for every $0\leq k\leq n$ and sequence of maps $f_1,f_2,f_3,\dots:S^k\to X$ that converges toward a point $x\in X$, the maps $f_m$ contract by a sequence of null-homotopies that…