# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Approximable dimension and acyclic resolutions

### Tom 152 / 1997

Fundamenta Mathematicae 152 (1997), 43-53 DOI: 10.4064/fm-152-1-43-53

#### Streszczenie

We establish the following characterization of the approximable dimension of the metric space X with respect to the commutative ring R with identity: $a-dim_R$ X ≤ n if and only if there exist a metric space Z of dimension at most n and a proper $UV^{n-1}$-mapping f:Z → X such that $\check H^n(f^-1}(x);R) = 0$ for all x ∈ X. As an application we obtain some fundamental results about the approximable dimension of metric spaces with respect to a commutative ring with identity, such as the subset theorem and the existence of a universal space. We also show that approximable dimension (with arbitrary coefficient group) is preserved under refinable mappings.

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