The minimal cardinality where the Reznichenko property fails

Boaz Tsaban

doi:10.1007/bf02786640

Home/Archive/arXiv

arXiv · math/0304024

The minimal cardinality where the Reznichenko property fails

Boaz Tsaban

View original source ↗Read source document ↗Export metadata ↓

Abstract

A topological space X$ has the Frechet-Urysohn property if for each subset A of X and each element x in the closure of A, there exists a countable sequence of elements of A which converges to x. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect all neighborhoods of x. In their paper, Kocinac and Scheepers conjecture that the minimal cardinality of a set X of real numbers such that C_p(X) does not have the weak Frechet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space). We prove the Kocinac-Scheepers conjecture by showing that if C_p(X) has the Reznichenko property, then a continuous image of X cannot be a subbase for a non-feeble filter on the natural numbers.

Explore related subjects

math.GN math.CA math.CO math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Boaz Tsaban. 2010-10-31. The minimal cardinality where the Reznichenko property fails. https://doi.org/10.1007/bf02786640

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sobriety of Scott topologies under countability conditions

In this paper, we focus on the sobriety of the Scott topology in countable case. Specifically, we show that: (1) every meet-continuous core-compact countable dcpo is sober with respect to the Scott topology; (2)every countable locally compact dcpo is sober endowed with the Scott topology; (3) the lattice of all open sets for the rational numbers space Q equipped with the Scott topology is not sober.

math.GN↗

arXiv2026

Exponentiable Objects and Function spaces in Lowen Fuzzy Topological Spaces

We study exponentiable objects and function spaces in the category of stratified Lowen fuzzy topological spaces over $\I=[0,1]$. Using the Lowen fuzzy Sierpiński object $\Sier$, which identifies $τ_X$ with $C(X,\Sier)$, we explicitly determine the largest splitting topology on this mapping set. Its open weights $Φ:τ_X\to\I$ are precisely those satisfying Scott continuity and a finite-tier compatibility condition induced by finite powers of $\Sier$. This yields an intrinsic characterization: $X$ is exponentiable if and only if every $μ\inτ_X$ satisfies \[ μ=\bigvee_{λ\triangleleftΦ} (\const{Φ(μ)}\wedgeλ), \qquad λ\triangleleftΦ \Longleftrightarrow \const{Φ(ν)}\wedgeλ\leqν \quad(ν\inτ_X). \] When this condition holds, $Y^X$ has underlying set $C(X,Y)$, with topology generated by $[Φ,v](f)=Φ(v\circ f)$. We also obtain a dual closed-set formulation and three applications. Exponentiability implies that $τ_X$ is a continuous lattice, although the converse fails. Moreover, a classical space $X$ is exponentiable exactly when its induced fuzzy space $ωX$ is exponentiable in the entire stratified Lowen category. Finally, Lowen compact, strongly fuzzy compact, and $N$-compact Hausdorff spaces are exponentiable.

math.GN↗

arXiv2026

Super calibers in topological spaces and topological hyperspaces

We study the notion of a super caliber of a topological space, which is closely related to the classical notion of caliber and has appeared in the literature under several different names. We collect and unify several known results and establish new results concerning the collections of super calibers of topological spaces and their hyperspaces. In particular, we investigate the relationship between the super calibers of a space $X$ and those of hyperspaces $\mathcal{H}(X)$ lying between $\mathrm{CL}(X)$ and $\mathcal{F}(X)$. For infinite metrizable spaces, we characterize several cases in which $\mathsf{SC}(X)$ and $\mathsf{SC}(\mathrm{CL}(X))$ differ and establish an independence result over \textsf{ZFC}; see Theorem~5.12.

math.GN↗