On infinite-dimensional manifold triples. (English) Zbl 0713.57010
Let \(Q=[-1,1]^{\omega}\), \(s=(-1,1)^{\omega}\), \(\Sigma =\{(x_ i)\in s:\sup | x_ i| <1\}\), and \(\sigma =\{(x_ i)\in s:\) \(x_ i=0\) for almost all \(i\}\). The authors study triples of spaces (X,M,N) that are locally homeomorphic, as triples, to either (Q,\(\Sigma\),\(\sigma\)) or (s,\(\Sigma\),\(\sigma\)). The main tools for this study come from an examination of the notion of cap set and f.d. cap set in the setting of pairs, and the main result is the equivalence of the following three conditions where X is a Q- (resp. s-) manifold and \(N\subset M\subset X:\) (1) the triple (X,M,N) is a (Q,\(\Sigma\),\(\sigma\))- (resp. (s,\(\Sigma\),\(\sigma\))-) manifold triple; (2) (M,N) is a cap- (resp. f.d. cap-) pair for X; (3) M is a strong cap set for the pair (X,N) and N is an f.d. cap set for X. Some applications of this main result are included.
Reviewer: R.Sher
MSC:
| 57N20 | Topology of infinite-dimensional manifolds |
Keywords:
Hilbert cube; Hilbert space; \({\mathbb{R}}^{\infty }\) Hilbert cube manifolds; \({\mathbb{R}}^{\infty }\)-manifolds; (Q,\(\Sigma \) ,\(\sigma \) )- manifold triple; cap-pairReferences:
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