Vol. 79, No. 1, 1978

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Homotopy properties of locally compact spaces at infinity-calmness and smoothness

Zvonko Cerin

Vol. 79 (1978), No. 1, 69–91

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.

Mathematical Subject Classification 2000
Primary: 55P55
Secondary: 57N25
Received: 1 June 1977
Published: 1 November 1978
Zvonko Cerin