Related papers: On weakening tightness to weak tightness

In this paper, authors prove that if $X$ is a weak Asplund space, then the space $X\times R$ is a weak Asplund space. Thus the author definitely answered an open problem raised by D.G. Larman and R.R. Phelps for 45 years ago (J. London.…

We consider special subclasses of the class of Lindel\"of Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $L\Sigma(\leq\kappa)$ if it…

The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for any neighborhood assignment $U:X\to \tau_X$ there is a set $A\in [X]^\kappa$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$.…

We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some $\kappa>\lambda\geq \aleph_0$, a continuous theory $T$ has a model with density character $\kappa$ which has a definable subset of density character…

Suppose that $\eta$ is a whole-plane space-filling SLE$_\kappa$ for $\kappa \in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $\eta(0) = 0$. For each $T > 0$ and $\kappa \in (4,8)$ we let…

The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the closed convex…

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $\kappa$ is a singular strong limit cardinal and…

Let $(\mathbb{X},d,\mu)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space…

The $G_\delta$-modification $X_\delta$ of a topological space $X$ is the space on the same underlying set generated by, i.e. having as a basis, the collection of all $G_\delta$ subsets of $X$. Bella and Spadaro recently investigated the…

We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when…

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence…

In this paper we introduce a notion of tightness for a family of nonlinear expectations and show that the tightness can be applied to obtain weak compactness in a framework of nonlinear expectation space. This criterion is very useful for…

In this article we extend the notion of orthogonal metric space to weak orthogonal metric space. Then we establish fixed point results for a mapping satisfying a more general contraction condition. Several nontrivial examples are given in…

We show that $\mathsf{PFA}$ implies that the tightness $t(X_\delta)$ of the $G_\delta$-modification of a Fr\'echet $\alpha_1$-space $X$ is at most $\omega_1$, while $\Box(\kappa)$ implies that there is a Fr\'echet $\alpha_1$-space with…

In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence.…

A well-known result of R. Pol states that a Banach space $X$ has property ($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of any convex set $C \subseteq B_{X^*}$ is actually in the weak*-closure of a countable…

A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of $X$ into $\mathbb{R}^\kappa$ the image $f(X)$ is compact. This notion generalizes pseudocompactness and gives a stratification of spaces lying…

This paper continues the investigation begun in arXiv:1906.05602 of extending the T1 theorem of David and Journ\'e, and optimal cancellation conditions, to more general weight pairs. The main additional tool developed here is a two weight…

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every…

The statement in the title solves a problem raised by T. Retta. We also present a variation of the result in terms of $[ \mu ,\kappa ]$-compactness.