arXiv subjects

Dušan Repovš

Publications and source records attributed to Dušan Repovš.

At least 91 records · Page 5Linked to original sources

arXiv2013

On 2-dimensional nonaspherical cell-like Peano continua: A simplified approach

We construct a functor $AC(-,-)$ from the category of path connected spaces $X$ with a base point $x$ to the category of simply connected spaces. The following are the main results of the paper: (i) If $X$ is a Peano continuum then $AC(X,x)$ is a cell-like Peano continuum; (ii) If $X$ is $n-$dimensional then $AC(X, x)$ is $(n+1)-$dimensional; and (iii) For a path connected space $X$, $π_1(X,x)$ is trivial if and only if $π_2(AC(X, x))$ is trivial. As a corollary, $AC(S^1, x)$ is a 2-dimensional nonaspherical cell-like Peano continuum.

math.GT↗

arXiv2013

On generalized 3-manifolds which are not homologically locally connected

We show that the classical example $X$ of a 3-dimensional generalized manifold constructed by van Kampen is another example of not homologically locally connected (i.e. not HLC) space. This space $X$ is not locally homeomorphic to any of the compact metrizable 3-dimensional manifolds constructed in our earlier paper which are not HLC spaces either.

math.GT↗

arXiv2012

The Geometry and Fundamental Groups of Solenoid Complements

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements.

math.GT↗

arXiv2012

$L$-fuzzy strongest postcondition predicate transformers as $L$-idempotent linear or affine operators between semimodules of monotonic predicates

For a completely distributive quantale $L$, $L$-fuzzy strongest postcondition predicate transformers are introduced, and it is shown that, under reasonable assumptions, they are linear or affine continuous mappings between continuous $L$-idempotent semimodules of $L$-fuzzy monotonic predicates.

math.CT↗

arXiv2012

On monoids of monotone injective partial self-maps of integers with cofinite domains and images

We study the semigroup $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ of monotone injective partial selfmaps of the set of integers having cofinite domain and image. We show that $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ is bisimple and all of its non-trivial semigroup homomorphisms are either isomorphisms or group homomorphisms. We also prove that every Baire topology $τ$ on $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ such that $(\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z}),τ)$ is a Hausdorff semitopological semigroup is discrete and we construct a non-discrete Hausdorff inverse semigroup topology $τ_W$ on $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$. We show that the discrete semigroup $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ cannot be embedded into some classes of compact-like topological semigroups and that its remainder under the closure in a topological semigroup $S$ is an ideal in $S$.

math.GR↗

arXiv2012

Nonexistence of linear operators extending Lipschitz (pseudo)metric

We present an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear extension operators for Lipschitz functions.

math.GN↗

arXiv2012

On Minc's sheltered middle path

This paper shows that a construction, which was introduced by Piotr Minc in connection with a problem that came from Helly type theorems and that allows to replace three PL-arcs with a "sheltered middle path", can in the case of general (non-PL) paths result in the topologist's sine curve.

math.GT↗

arXiv2012

Hyperspaces of max-plus convex subsets of powers of the real line

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^τ$ of the real line. We show that the corresponding spaces are AR's if and only if $τ\leω_1$.

math.GN↗

arXiv2012

On the homology of the Harmonic Archipelago

We calculate the singular homology and Čech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic.

math.GT↗

arXiv2012

A two-parameter control for contractive-like multivalued mappings

We propose a general approach to defining a contractive-like multivalued mappings $F$ which avoids any use of the Hausdorff distance between the sets $F(x)$ and $F(y)$. Various fixed point theorems are proved under a two-parameter control of the distance function $d_{F}(x)=dist(x,F(x))$ between a point $x \in X$ and the value $F(x) ßX$. Here, both parameters are numerical functions. The first one $\a\,:[0,+ı)\rightarrow [1,+ı)$ controls the distance between $x$ and some appropriate point $y \in F(x)$ in comparison with $d_{F}(x)$, whereas the second one $\b\,:[0,+ı)\rightarrow [0,1)$ estimates $d_{F}(y)$ with respect to $d(x,y)$. It appears that the well harmonized relations between $\a$ and $\b$ are sufficient for the existence of fixed points of $F$. Our results generalize several known fixed-point theorems.

math.GN↗

arXiv2012

On Murayama's theorem on extensor properties of G-spaces of given orbit types

We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.

math.GT↗

arXiv2012

Survey on the Generalized R. L. Moore Problem

We give an updated extended survey of results related to the celebrated unsolved generalized R. L. Moore problem. In particular, we address the problem of characterizing codimension one manifold factors, i.e. spaces $X$ having the property that $X \times \mathbb{R}$ is a topological manifold. A main part of the paper is devoted to many efficient general position techniques, which have been used to solve special cases of this problem.

math.GT↗

arXiv2011

On Pliś metric on the space of strictly convex compacta

We consider a certain metric on the space of all convex compacta in $\R^{n}$, introduced by A. Pliś. The set of strictly convex compacta is a complete metric subspace of the metric space of convex compacta with respect to this metric. We present some applications of this metric to the problems of set-valued analysis, in particular we estimate the distance between two compact sets with respect to this metric and to the Hausdorff metric.

math.GN↗

arXiv2011

Topics in uniform continuity

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous.

math.GN↗

arXiv2011

Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $τ$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),τ)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),τ)$ in a topological semigroup.

math.GN↗

arXiv2011

The Banach-Mazur compactum is the Alexandroff compactification of a Hilbert cube manifold

We prove that for every $n>2$, the Banach-Mazur compactum Q(n) is the compactification of a Hilbert cube manifold by the Euclidean point. For $n=2$ this result was proved earlier.

math.GT↗

arXiv2011

On the singular homology of one class of simply-connected cell-like spaces

In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called {\sl Snake space}. In the sequel we introduced the functor $SC(-,-)$ defined on the category of all spaces with base points and continuous mappings. For the circle $S^1$, the space $SC(S^1, \ast)$ is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces $SC(Z, \ast)$ for any path-connected compact spaces $Z$.

math.GN↗

arXiv2011

Idempotent convexity and algebras for the capacity monad and its submonads

Idempotent analogues of convexity are introduced. It is proved that the category of algebras for the capacity monad in the category of compacta is isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and their biaffine maps. It is also shown that the category of algebras for the monad of sup-measures ($(\max,\min)$-idempotent measures) is isomorphic to the category of $(\max,\min)$-idempotent convex compacta and their affine maps.

math.CT↗