### Abstract

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 Ȟ^{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.

Original language | English |
---|---|

Pages (from-to) | 43-53 |

Number of pages | 11 |

Journal | Fundamenta Mathematicae |

Volume | 152 |

Issue number | 1 |

Publication status | Published - 1997 |

Externally published | Yes |

### Fingerprint

### Keywords

- Acyclic resolution
- Approximable dimension
- Cohomological dimension
- Refinable mapping
- Universal space
- Uv -resolution

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*152*(1), 43-53.

**Approximable dimension and acyclic resolutions.** / Koyama, Akira; Sher, R. B.

Research output: Contribution to journal › Article

*Fundamenta Mathematicae*, vol. 152, no. 1, pp. 43-53.

}

TY - JOUR

T1 - Approximable dimension and acyclic resolutions

AU - Koyama, Akira

AU - Sher, R. B.

PY - 1997

Y1 - 1997

N2 - We establish the following characterization of the approximable dimension of the metric space X with respect to the commutative ring R with identity: a-dimR X ≤ n if and only if there exist a metric space Z of dimension at most n and a proper UVn-1-mapping f : Z → X such that Ȟn(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.

AB - We establish the following characterization of the approximable dimension of the metric space X with respect to the commutative ring R with identity: a-dimR X ≤ n if and only if there exist a metric space Z of dimension at most n and a proper UVn-1-mapping f : Z → X such that Ȟn(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.

KW - Acyclic resolution

KW - Approximable dimension

KW - Cohomological dimension

KW - Refinable mapping

KW - Universal space

KW - Uv -resolution

UR - http://www.scopus.com/inward/record.url?scp=0031461657&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031461657&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031461657

VL - 152

SP - 43

EP - 53

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 1

ER -