Embeddings in the Fell and Wijsman topologies

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].