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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.

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    Imagine an octopus with infinitely long arms, and then let it have infinitely many.2011-08-15
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    @scoobs: What's your actual question? Do you want an explanation of what an ‘end’ is?2011-08-15
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    @Harry: since the OP wants infinitely many, perhaps it is easier to start with a hedgehog or a porcupine. `:)` (But be careful not to confused it with the Cartwright-Littlewood hedgehog.)2011-08-15
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    Instead of a hedgehog with infinitely many spines, perhaps it might be easier to imagine a tree with infinitely long branches that keep splitting as they extend to infinity.2011-08-15
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    Hi 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

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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.

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http://www.math.indiana.edu/gallery/minimalSurface.phtml

has infinitely many ends

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    Thanks, Will, great graphics!2011-08-18