Related papers: Finite Sets with Fake Observable Cardinality

Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {f(x_1,...,x_n): x_1\in A_1,...,x_n\in A_n, and x_i\not=x_j…

We prove that the set of all endpoints of the Julia set of $f(z)=\exp(z)-1$ which escape to infinity under iteration of $f$ is not homeomorphic to the rational Hilbert space $\mathfrak E$. As a corollary, we show that the set of all points…

It is consistent with ZF + DC that there exists an ultrafilter $U$ on $\omega$ such that two infinite ultraproducts of finite sets, $\prod A_n / U$ and $\prod B_n / U$, have the same cardinality if and only if $0 < \lim_U |A_n|/|B_n| <…

A space $X$ is said to be $\kappa$-resolvable (resp. almost $\kappa$-resolvable) if it contains $\kappa$ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets). $X$ is maximally resolvable iff…

A simple criterion for a functor to be finitary is presented: we call $F$ finitely bounded if for all objects $X$ every finitely generated subobject of $FX$ factorizes through the $F$-image of a finitely generated subobject of $X$. This is…

A finite set X in the d-dimensional Euclidean space is called an s-distance set if the set of Euclidean distances between any two distinct points of X has size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance set…

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold must have at least as many fixed points as the minimal number of critical points of a smooth function on the manifold. It is well…

We give a new and elementary proof showing that a homeomorphism of a compact metric space is positively expansive if and only if the space is finite.

It is an interesting, maybe surprising, fact that different dense subspaces of even "nice" topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological…

We consider point sets in the $m$-dimensional affine space $\mathbb{F}_q^m$ where each squared Euclidean distance of two points is a square in $\mathbb{F}_q$. It turns out that the situation in $\mathbb{F}_q^m$ is rather similar to the one…

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper,…

For a given continuum $X$ and a natural number $n,$ we consider the hyperspace $F_n(X)$ of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. In this paper we show that if $X$ is a dendrite whose set of…

Let $\eta_{g}(n) $ be the smallest cardinality that $A\subseteq {\mathbb Z}$ can have if $A$ is a $g$-difference basis for $[n]$ (i.e, if, for each $x\in [n]$, there are {\em at least} $g$ solutions to $a_{1}-a_{2}=x$ ). We prove that the…

Let $f$ be an $R$-closed homeomorphism on a connected orientable closed surface $M$. In this paper, we show that If $M$ has genus more than one, then each minimal set is either a periodic orbit or an extension of a Cantor set. If $M =…

If $\mathcal P$ is a family of filters over some set $I$, a topological space $X$ is \emph{sequencewise $\mathcal P$-\brfrt compact} if, for every $I$-indexed sequence of elements of $X$, there is $F \in \mathcal P$ such that the sequence…

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $\rho$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $\beta = \sup \big\{ \rho(x, y): \{x, y \} \subset \mathcal…

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 a recent paper \cite{T} the fact that a class of locally compact metric spaces $X$, among which are Euclidean spaces, are not homemorphic to their punctured version $X\men\{p\}$, was given an interesting new proof which does not use…

A topological space ${\mathcal X}$ is reversible iff each continuous bijection (condensation) $f: {\mathcal X} \rightarrow {\mathcal X}$ is a homeomorphism; weakly reversible iff whenever ${\mathcal Y}$ is a space and there are…

A space $X$ is said to be "cellular-Lindel\"of" if for every cellular family $\mathcal{U}$ there is a Lindel\"of subspace $L$ of $X$ which meets every element of $\mathcal{U}$. Cellular-Lindel\"of spaces generalize both Lindel\"of spaces…