Open problems on countable dense homogeneity

On some generated soft topological spaces and soft homogeneity

“…A topological space true(X,ℑtrue) is homogeneous if for all x,y∈X, there is a homeomorphism f:(X,ℑ)false⟶(X,ℑ) such that ffalse(xfalse)goodbreakafter=y. Since homogeneity concepts are of importance in general topology and still a hot area of research, as appears in [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], we see that it is suitable to extend the homogeneity concept to include soft topological spaces. One of our main goals of the present work is to show how the definition of homogeneity in ordinary topological spaces can be modified in order to define its extension in soft topological spaces.…”

Countable dense homogeneity and the Cantor set

“…A space X is countable dense homogeneous (CDH, henceforth) if X is separable and every time D, E ⊂ X are countable dense subsets, there is a homeomorphism h : X → X such that h[D] = E. Among examples of CDH spaces we have the Euclidean spaces, the Hilbert cube and the Cantor set. For updated surveys on CDH spaces, see sections 14, 15 and 16 of [2] and the more recent [9].…”

Countable dense homogeneity of function spaces

“…A space X is countable dense homogeneous (CDH, henceforth) if X is separable and every time D, E ⊂ X are countable dense subsets, there is a homeomorphism h : X → X such that h[D] = E. Among examples of CDH spaces we have the Euclidean spaces, the Hilbert cube and the Cantor set. For updated surveys on CDH spaces, see sections 14, 15 and 16 of [2]; and [10].…”