Embeddings in the Fell and Wijsman topologies

TL;DR

This paper investigates embeddings of complex topological structures into hyperspaces equipped with Fell and Wijsman topologies, revealing conditions under which certain uncountable product spaces embed, thus advancing understanding of hyperspace topology.

Contribution

It establishes new embedding results for uncountable product spaces into hyperspaces with Fell and Wijsman topologies under specific topological conditions.

Findings

$( au_F)$-hyperspaces contain $( ext{omega}_1+1)^{ ext{omega}}$ and $( ext{omega}_1+1)^{ ext{omega}_1}$ if $X$ has a closed uncountable discrete subspace.

$( au_{w(d)})$-hyperspaces embed these spaces if $X$ is non-separable and perfect.

Provides partial answers to existing questions about hyperspace embeddings.

Abstract

It is shown that if a $T_2$ topological space $X$ contains a closed uncountable discrete subspace, then the spaces $(\omega_1 + 1)^{\omega}$ and $(\omega_1 + 1)^{\omega_1}$ embed into $(CL(X),\tau_F)$, the hyperspace of nonempty closed subsets of $X$ equipped with the Fell topology. If $(X,d)$ is a non-separable perfect topological space, then $(\omega_1 + 1)^{\omega}$ and $(\omega_1 + 1)^{\omega_1}$ embed into $(CL(X),\tau_{w(d)})$, the hyperspace of nonempty closed subsets of $X$ equipped with the Wijsman topology, giving a partial answer to the Question 3.4 in [CJ].