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A three dimensional prime end theory. (English) Zbl 0873.57012
Summary: Prime end theory is essentially a compactification theory for simply connected, bounded domains, $$U$$, in $$E^2$$, or simply connected domains in $$S^2$$ with nondegenerate complement. The planar case was originally due to C. Caratheodory [Math. Ann. 73, 323-370 (1913; JFM 44.0757.02)] and was later generalized to the sphere by H. D. Ursell and L. C. Young [Mem. Am. Math. Soc. 3 (1951; Zbl 0043.16902)], and to arbitrary two-manifolds by J. N. Mather [Sel. Stud.: Physics-Astrophysics, Mathematics, History of Science, Vol. dedic. A. Einstein, 225-255 (1982; Zbl 0506.57005)]. There are many applications of the two-dimensional theory, including applications to fixed point problems, embedding problems, and homeomorphism (group) actions.
Several constructions of a three-dimensional topological prime end theory appear in the literature, including work by B. Kaufmann [Math. Ann. 103, 70-144 (1930; JFM 56.0848.01)], S. Mazurkiewicz [Fundam. Math. 33, 177-228 (1945; Zbl 0060.40009)], and D. B. A. Epstein [Proc. Lond. Math. Soc., III. Ser. 42, 385-414 (1981; Zbl 0491.30027)].
In this paper, the authors develop a simple three-dimensional prime end theory for certain open subsets of Euclidean three-space. It includes conditions focusing on an “Induced Homeomorphism Theorem”, which, the authors believe, provides the necessary ingredient for applications.

##### MSC:
 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010)
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