Fake boundary sets in the Hilbert cube

Author:
Philip L. Bowers

Journal:
Proc. Amer. Math. Soc. **93** (1985), 121-127

MSC:
Primary 57N20; Secondary 54F35

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766541-4

MathSciNet review:
766541

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Abstract: For each positive integer , a --set in the Hilbert cube is constructed whose complement is not homeomorphic to the pseudointerior of the Hilbert cube though and satisfy: (i) every compact subset of is a -set in ; (ii) is homeomorphic to ; (iii) admits small maps ; (iv) satisfies the discrete -cells property; and (v) is locally -connected in . It is shown that does not satisfy the discrete -cells property and thus is not a boundary set, that is, is not homeomorphic to . These examples build upon an example of Anderson, Curtis, and van Mill of a fake boundary set that satisfies (i)-(iv) for . Their example is not a boundary set since it fails to be locally continuum-connected. The examples constructed herein show that there is a hierarchy of fake boundary sets satisfying (i)-(iv) that satisfy higher and higher orders of a strong form of local connectivity (v).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0766541-4

Keywords:
Boundary set in the Hilbert cube,
discrete approximation property,
discrete -cells property,
locally -connected in

Article copyright:
© Copyright 1985
American Mathematical Society