We define two properties of
noncompact locally compact spaces called 𝒞-calmness at ∞ and (𝒞,𝒟)-smoothness at
∞ for arbitrary classes of topological spaces 𝒞 and 𝒟. A number of theorems and
examples concerning these properties are given. By considering complements of
Z-sets in the Hilbert cube from them we get three new shape invariant conditions for
compact metric spaces named calmness, n-calmness, and n-smoothness. Calmness is
a movability type condition while n-smoothness implies that (and under some
additional assumptions is also implied by) the k-th shape pro-group of a compactum
in question is trivial, for all k > n.
