On \(C\)-embeddedness of hyperspaces (Q388815)

\(\mathcal{K}(X)\) is the hyperspace of non-empty closed and compact sets and \(\mathcal{C L}(X)\) is the hyperspace of non-empty closed sets, both endowed with the Vietoris topology. The paper contributes to the solution of the following question: Under which conditions is \(\mathcal{K}(X)\) \(C^{*}\)-embedded in \(\mathcal{C L}(X)\)? In the paper the authors restrict the question to ordinal spaces \(X= [0,\gamma)\). The main result of the paper formulates an answer to the question: For an ordinal space \(X= [0,\gamma)\) the space \(\mathcal{K}([0,\gamma))\) is \(C^{*}\)-embedded in \(\mathcal{C L}([0,\gamma))\) if and only if \(cof(\gamma) \not = \omega\).