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Proceedings of the American Mathematical Society

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

 
 

 

Tree-like continua and cellularity


Author: R. Richard Summerhill
Journal: Proc. Amer. Math. Soc. 26 (1970), 201-205
MSC: Primary 54.55
DOI: https://doi.org/10.1090/S0002-9939-1970-0275375-8
MathSciNet review: 0275375
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Abstract: In this paper the equivalence of tree-like and cellular is proved for $ 1$-dimensional continua in $ {E^n}$. More precisely, if $ X$ is a tree-like continuum, then the collection of all embeddings $ h:X \to {E^n},n \geqq 3$, such that $ h[X]$ is cellular in $ {E^n}$ is a dense $ {G_\delta }$-subset of the collection of all maps from $ X$ into $ {E^n}$. Conversely, if $ X$ is a $ 1$-dimensional cellular subset of $ {E^n}$, then $ X$ is a tree-like continuum.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0275375-8
Keywords: Cellularity, continua, dimension, tree-like, UV-properties
Article copyright: © Copyright 1970 American Mathematical Society

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