Very slight edit inserted in Dan's explanation, which I'm pretty sure he'll agree with: On Tue, Feb 25, 2014 at 10:28 PM, Dan Asimov <dasimov@earthlink.net> wrote:
To a topologist, all Cantor sets are, by definition, homeomorphic to the archetypal middle-third one, and so to each other.
Two points of a topological space are said to be in the same connected component if there do not exist two disjoint open sets
... *whose union is the whole space *...
each containing one of the two points.
From this definition it's easy to check that the connected components of the (any) Cantor set are its individual points. Hence the Cantor set is called "totally disconnected".
As Allan points out, every point of the Cantor set is a limit point of the set, i.e., no point is isolated. Yet this is consistent with its being totally disconnected.
--Dan
On Feb 25, 2014, at 4:01 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I'm not even sure if the archetypical Cantor set should be described as a "collection of zero-dimensional points", except in the sense that *every * point set is such a collection. No element of the Cantor set is isolated, in the sense that every neighborhood of a point in the Cantor set also contains other points in the set.
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