I was reading an article that mentioned "a connected surface in 3D space with $\infty$ many ends (in the topologocal sense)". I have read the wiki page on "ends" but couldn't make much sense of it, not to mention being able to come up/ visualise such a surface! Help would be very much appreciated.
3D surface and topology
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0Hi everyone, sorry about the delay in replying. My internet was down. I would appreciate an explanation of the definition of an "end". And thanks for the other suggestions too! So the afore described surface is the "boundary" of an infinitely branching tree or a hedgehog with infinitely many spikey things? – 2011-08-16
2 Answers
Let $\pi_0 (Y)$ denote the set of connected components of a topological space $Y$. (This usually means the set of path-connected components of $Y$, but for our purposes here this is more convenient.) The set of ends of $X$ is the subset $E$ of the cartesian product $\prod_K \pi_0 (X \setminus K)$, where $K$ varies over all the compact subsets of $X$, defined by the following condition: $(U_\bullet) \in E$ just if for every pair of compact subsets $K$ and K', there is a compact subset K'' such that K'' \subseteq K \cap K' and U_K \cup U_{K'} \subseteq U_{K''}.
Intuitively, an end of $X$ is a connected component of $X$ ‘at infinity’. For example, you may wish to verify that $\mathbb{R}$ has two ends, $\mathbb{R}^2$ has one end, and $[0, 1]$ has no ends. As suggested in the comments, there are many intuitive objects which have infinitely many ends. A tree which bifurcates forever would be my favourite.
http://www.math.indiana.edu/gallery/minimalSurface.phtml
has infinitely many ends
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0Thanks, Will, great graphics! – 2011-08-18