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 | References | Similar Articles | Additional Information
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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Additional Information
Keywords:
Cellularity,
continua,
dimension,
tree-like,
<I>UV</I>-properties
Article copyright:
© Copyright 1970
American Mathematical Society