Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Sets accessible at each point only by wild arcs


Author: Gary Glenn Miller
Journal: Proc. Amer. Math. Soc. 31 (1972), 583-590
DOI: https://doi.org/10.1090/S0002-9939-1972-0284992-2
MathSciNet review: 0284992
Full-text PDF Free Access

Abstract | References | Additional Information

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.


References [Enhancements On Off] (What's this?)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0284992-2
Keywords: Arcwise accessible, wild arc, tame $ 2$-complex, arcwise connected, locally connected, dense subset of $ {E^3}$
Article copyright: © Copyright 1972 American Mathematical Society