Modulo arithmetic of function spaces: Subset hyperspaces as quotients of function spaces
Earnest Akofor
TL;DR
This work studies representing subset hyperspaces $\mathcal{J}\subset Cl(X)$ as quotient spaces of function spaces $\mathcal{F}\subset X^Y$ via the unordering map $q:f\mapsto cl_X(f(Y))$. It shows that hyperspace topologies that extend the Vietoris topology arise as quotients of suitably chosen function-space topologies, and constructs $\tau_\pi$ on $\mathcal{F}$ whose quotient $\tau_{\pi q}$ contains $\tau_v$; in particular, it proves the metrizability of $K_Y(X)$ through the Hausdorff metric $d_H$ under metrizable $X$, and provides concrete quotient realizations (e.g., via $\tau_{cc}$) that realize the Vietoris topology as a quotient. The paper also establishes the existence of $\tau_p$-compatible lifts of hyperspace topologies (termed swrc-potent lifts), enabling quotients of $\mathcal{F}$ to realize given $\tau_v$-compatible topologies on $Cl_Y(X)$; results are complemented by a discussion of when such quotients yield metrizability and by open questions on Lipschitz retracts and path representations. Overall, the work builds a bridge between function-space convergence notions and hypertopology, offering a unified framework for analyzing and metrizing subset hyperspaces with quotient constructions.
Abstract
Let $X$ be a (topological) space and $Cl(X)$ the collection of nonempty closed subsets of $X$. Given a topology on $Cl(X)$, making $Cl(X)$ a space, a (subset) hyperspace of $X$ is a subspace $\mathcal{J}\subset Cl(X)$ with an embedding $X\hookrightarrow\mathcal{J}$, $x\mapsto\{x\}$. In this note, we characterize certain hyperspaces $\mathcal{J}\subset Cl(X)$ as explicit quotient spaces of function spaces $\mathcal{F}\subset X^Y$ and discuss metrization of associated compact-subset hyperspaces in this setting. In particular, we find that any hyperspace topology containing the Vietoris topology is a quotient of a function space topology containing the topology of pointwise convergence.