**ERIC Number:**ED518790

**Record Type:**Non-Journal

**Publication Date:**2010

**Pages:**119

**Abstractor:**As Provided

**Reference Count:**0

**ISBN:**ISBN-978-1-1242-1629-4

**ISSN:**N/A

A Thin Codimension-One Decomposition of the Hilbert Cube

Phon-On, Aniruth

ProQuest LLC, Ph.D. Dissertation, Oregon State University

For cell-like upper semicontinuous (usc) decompositions "G" of finite dimensional manifolds "M", the decomposition space "M/G" turns out to be an ANR provided "M/G" is finite dimensional ([Dav07], page 129). Furthermore, if "M/G" is finite dimensional and has the Disjoint Disks Property (DDP), then "M/G" is homeomorphic to "M" ([Dav07], page 181). For an infinite dimensional "M" modeled on I[infinity], we can construct cell-like usc decompositions "G" associated with defining sequences. But it is more complicated to check whether "M/G" is an ANR. We need an additional special property of the defining sequence. To check whether or not "M/G" is homeomorphic to "M" is even more difficult. We need "M/G" to be an ANR which has the DDP and which also satisfies the Disjoint Cech Carriers Property. We give a specific cell-like decomposition "X" of the Hilbert Cube "Q" with the following properties: The nonmanifold part "N" of "X" is complicated in the sense that it is homeomorphic to a Hilbert Cube of codimension 1 in "Q". "X" is still a factor of "Q" because X x I[superscript 2] [congruent with] "Q". If "A" is any closed subspace of "N" of codimension [greater than or equal to] 1 in "N", then the decomposition of "Q" over "A" is homeomorphic to "Q". In particular, the nonmanifold nature of "X" is not detectable by examining closed subsets of codimension [greater than or equal to] 1. This example is produced by combining mixing techniques for producing a nonmanifold space whose nonmanifold part is a Cantor set, with decompositions arising from a generalized Cantor function. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]

Descriptors: Mathematics Instruction, Mathematical Concepts, Geometric Concepts, Validity, Mathematical Logic

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**Publication Type:**Dissertations/Theses - Doctoral Dissertations

**Education Level:**N/A

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**Language:**English

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