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].