Search results for "Mathematics::General Topology"

Space-filling vs. Luzin's condition (N)

2013

Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the Hausdorff measure of the corresponding dimension. We show for quite general metric spaces that for any measurable surjection from the first onto the second space, there is a set of measure zero that is mapped to a set of positive measure (both measures are the Hausdorff measures corresponding to the Hausdorff dimension of the first space). We also study more general situations where the measures on the two metric spaces are not necessarily the same and not ne…

Fixed Points for Multivalued Convex Contractions on Nadler Sense Types in a Geodesic Metric Space

2019

In 1969, based on the concept of the Hausdorff metric, Nadler Jr. introduced the notion of multivalued contractions. He demonstrated that, in a complete metric space, a multivalued contraction possesses a fixed point. Later on, Nadler&rsquo

Free sequences and the tightness of pseudoradial spaces

2019

Let F(X) be the supremum of cardinalities of free sequences in X. We prove that the radial character of every Lindelof Hausdorff almost radial space X and the set-tightness of every Lindelof Hausdorff space are always bounded above by F(X). We then improve a result of Dow, Juhasz, Soukup, Szentmiklossy and Weiss by proving that if X is a Lindelof Hausdorff space, and $$X_\delta $$ denotes the $$G_\delta $$ topology on X then $$t(X_\delta ) \le 2^{t(X)}$$ . Finally, we exploit this to prove that if X is a Lindelof Hausdorff pseudoradial space then $$F(X_\delta ) \le 2^{F(X)}$$ .

Relative principal congruences in congruence-modular quasivarieties

1998

The problem of definability of relative principal congruences in relatively congruence modular (RCM) quasivarieties is investigated. The RCM quasivarieties are characterized in terms of parameterized families of finite sets of pairs of terms which define relative principal congruences.

Extended Natural Numbers and Counters

2020

Summary This article introduces extended natural numbers, i.e. the set ℕ ∪ {+∞}, in Mizar [4], [3] and formalizes a way to list a cardinal numbers of cardinals. Both concepts have applications in graph theory.

Visible parts and dimensions

2003

We study the visible parts of subsets of n-dimensional Euclidean space: a point a of a compact set A is visible from an affine subspace K of n, if the line segment joining PK(a) to a only intersects A at a (here PK denotes projection onto K). The set of all such points visible from a given subspace K is called the visible part of A from K. We prove that if the Hausdorff dimension of a compact set is at most n−1, then the Hausdorff dimension of a visible part is almost surely equal to the Hausdorff dimension of the set. On the other hand, provided that the set has Hausdorff dimension larger than n−1, we have the almost sure lower bound n−1 for the Hausdorff dimensions of visible parts. We al…

Fixed point results for Gm-Meir-Keeler contractive and G-(α,ψ)-Meir-Keeler contractive mappings

Pseudo-path connected homogeneous continua

2015

Abstract The main result of this paper states that every homogeneous pseudo-path connected continuum is weakly chainable, or equivalently, every homogeneous continuum connected by continuous images of the pseudo-arc is itself a continuous image of the pseudo-arc. We notice that even though there exist homogeneous path connected continua that are not continuous images of an arc (Prajs, 2002), they all are continuous images of the pseudo-arc.

Cardinal invariants of cellular Lindelof spaces

2018

A space X is said to be cellular-Lindelof if for every cellular family $$\mathcal {U}$$ there is a Lindelof subspace L of X which meets every element of $$\mathcal {U}$$ . Cellular-Lindelof spaces generalize both Lindelof spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelof monotonically normal space is Lindelof and that every cellular-Lindelof space with a regular $$G_\delta $$ -diagonal has cardinality at most $$2^\mathfrak {c}$$ . We also prove that every normal cellular-Lindelof first-countable space has cardinality at most continuum under $$2^{<\mathfrak {c}}=\mathfrak {c}$$ and that every normal cellular-Lindel…

IFS attractors and Cantor sets

2006

Abstract We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R 3 such that every homeomorphism f of R 3 which preserves K coincides with the identity on K.