Tightness type properties of spaces of quasicontinuous functions

TL;DR

This work investigates tightness-type invariants in spaces of quasicontinuous functions $Q_p(X,Y)$ under pointwise convergence, addressing whether such invariants are determined by the target space $Y$ or the domain $X$. It develops a framework combining approximation by continuous functions, domain-approximation (DA) families, and a flattening lemma based on hedgehog spaces to reduce questions to a universal target $\mathbb{H}$. The main results establish equivalences: for metrizable $X$, $t(Q_p(X,Y))=\omega$ for all metrizable $Y$ if and only if $X$ is $K_{\Omega}$-Lindelöf; analogous characterizations exist for $vet$ and $vet_1$ via the selection principles $S_{fin}(K_{\Omega},K_{\Omega})$ and $S_1(K_{\Omega},K_{\Omega})$. These findings resolve the open metrizable-case problem of Osipov by showing that countable tightness alone cannot distinguish points in $Q_p(X,Y)$ when $Y$ is metrizable, and they provide a robust methodological template for related tightness questions in function spaces.

Abstract

Using approximation by continuous functions we prove the following statements to types of tightness in a space $Q_p(X, \mathbb{R})$ of all quasicontinuous real-valued functions with the topology $τ_p$ of pointwise convergence: the countability of tightness (fan-tightness, strong fan-tightness) at a point $f$ of space $Q_p(X, \mathbb{R})$ implies the countability of tightness (fan-tightness, strong fan-tightness) of space $Q_p(X,Y)$ of all quasicontinuous functions from $X$ into any non-one-point metrizable space $Y$. This result is the answer to the open question in the class of metrizable spaces.

Theorems & Definitions (28)

• Definition 2.1
• Theorem 3.1
• Proposition 3.2
• proof
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof
• Corollary 4.4
• Definition 5.1