On stratifiability of mapping spaces (Q1575039)

Answering (in the negative and partially) a question by \textit{C. R. Borges} [General Topol. Appl. 1, 79-84 (1971; Zbl 0211.54502)] the authors construct a space \(X\) determined by compact metrizable subspaces such that the space \(C_k(S,X)\) of continuous mappings from the convergent sequence \(S\) (with the limit point) into \(X\) with the compact-open topology is not stratifiable. The same space \(X\) serves as an example of a space for which the space \(\mathcal K(X)\) of all non-empty compact subspaces of \(X\) with the Vietoris topology is not stratifiable. However, if \(M\) is a metrizable compactum and \(\mathcal K(Y)\) is stratifiable, then \(C_k(M,Y)\) is stratifiable.