Sets accessible at each point only by wild arcs

Miller, Gary Glenn

doi:10.1090/S0002-9939-1972-0284992-2

Sets accessible at each point only by wild arcs
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by Gary Glenn Miller

Proc. Amer. Math. Soc. 31 (1972), 583-590

DOI: https://doi.org/10.1090/S0002-9939-1972-0284992-2

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Abstract:

A positional pathology as described in the title is shown to occur in three-space. We construct an arcwise accessible point set $M$ such that each arc to $M$ is locally knotted at uncountably many points. In addition we give examples of connected, locally connected point sets $S$ and $T$ which are accessible at each point only by wild arcs and tame arcs respectively. The point set $M$ has countably many components, and each of these is a tame finite $2$-complex. Moreover ${E^3} - M$ is locally connected and arcwise connected.

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Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 583-590
DOI: https://doi.org/10.1090/S0002-9939-1972-0284992-2
MathSciNet review: 0284992