#### Date of Award

6-1986

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

John J. Walsh

#### Committee Members

Robert J. Daverman, Kwangh Jeon

#### Abstract

Recently, R.J. Daverman and J.J. Walsh modified an example due to J. Taylor to obtain an example of a cell-like map from a compactum with non-trivial shape onto the Hilbert cube Q such that the non-degeneracy set is contained in the countable union of finite dimensional closed subsets of Q. Previously, G. Kozlowski proved that a cell-like map f: X' → X from a compact ANR X' onto a metric space X is a hereditary shape equivalence if there exists a sequence {B_{n}}∞_{n=1} of finite dimensional closed subsets of X such that the non-degeneracy set is contained in ∞U_{n=1} B_{n} and {f^{-1} (B_{n})}∞_{n=1} forms a pairwise disjoint null-sequence. Here we raise two questions, which we show are equivalent. First: Is a cell-like map f: X' → X a hereditary shape equivalence if there exists a sequence {B_{n}}∞_{n=1} of finite dimensional closed subsets of X such that U_{n≠m} (B_{n} ∩ B_{m}) has a strong transfinite dimension and the non-degeneracy set is contained in ∞U_{n=1} B_{n} ? Even though we are not able to answer these questions, we give affirmative answers to the questions for special cases, and , furthermore, we are able to extend the aforementioned result of Kozlowski's. Also, we attempt to extend certain analyses of cell-like maps, which are proper by definition, to (non-proper) UV∞-maps. We prove that for a UV∞-map f: X' → X from an ANR X' to a metric space X the following are equivalent: (1) X is an ANR; (2) f is a hereditary homotopy equivalence; (3) f is a hereditary shape equivalence; (4) f is a fine homotopy equivalence. Since UV∞-maps are generally not onto, the notion of a "hereditary shape equivalence" is a variation of that for cell-like maps, though it agrees with, say, Kozlowski's for cell-like maps.

#### Recommended Citation

Choi, Jung-In Kang, "Mappings of ANR's Whose Images are ANR's. " PhD diss., University of Tennessee, 1986.

https://trace.tennessee.edu/utk_graddiss/2996