Cell-like maps that are shape equivalences

Choi, Jung-In K.

doi:10.1090/S0002-9939-1990-1038759-9

Cell-like maps that are shape equivalences
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by Jung-In K. Choi PDF

Proc. Amer. Math. Soc. 108 (1990), 1011-1018 Request permission

Abstract:

Let $f:X’ \to X$ be a cell-like map between metric spaces and set ${N_f} = \{ x \in X:{f^{ - 1}}(x) \ne {\text {point\} }}$. Even if ${N_f} \subset \bigcup \nolimits _{n = 1}^\infty {{B_n}}$, where each ${B_n}$ is closed and each $f|{f^{ - 1}}({B_n}):{f^{ - 1}}({B_n}) \to {B_n}$ is hereditary shape equivalence, $f$ may not be a hereditary shape equivalence. Conditions are placed on the ${B_n}$ ’s to assure that $f$ is a hereditary shape equivalence. For example, if ${N_f} \subset \bigcup \nolimits _{n = 1}^\infty {{B_n}}$, where ${B_n}$ is closed for each $n = 1,2, \ldots ,f|{f^{ - 1}}({B_n}):{f^{ - 1}}({B_n}) \to {B_n}$ is a hereditary shape equivalence, and ${B_n}$ has arbitrary small neighborhoods whose boundaries miss $\bigcup \nolimits _{i = 1}^\infty {{B_i}}$ then $f$ is a hereditary shape equivalence. An immediate consequence is that if $\{ {B_n}\} _{n = 1}^\infty$ is a pairwise disjoint null-sequence and each $f|{f^{ - 1}}({B_n})$ is a hereditary shape equivalence, then $f$ is a hereditary shape equivalence. Previously G. Kozlowski showed that if $\{ {f^{ - 1}}({B_n})\} _{n = 1}^\infty$ is a pairwise disjoint null-sequence and each $f|{f^{ - 1}}({B_n})$ is a hereditary shape equivalence, then $f$ is a hereditary shape equivalence, which can be obtained as an immediate corollary of one of our results.

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