TITLE: Basic Topology: Need example
QUESTION [5 upvotes]: I need help finding an example of an infinitely uncountable set that contains only isolated points.  I so far have been completely unsuccessful and have only found uncountable set with no isolated points.  Thanks for your help.

REPLY [5 votes]: There is no example as a subset of $\mathbb R^d$ (with the standard topology), or in any other separable metric space.
Each of your uncountably many points would, by definition, be surrounded by a ball that contains no other points in the set. A priori these balls may intersect, but you can avoid that by halving the radius of each of them. Then $\mathbb R^d$ would contain uncountably many disjoint balls. But that is impossible because each of the balls contains a point with rational coordinates, and there are only countably many of those to go around!
On the other hand, if you don't need your set to live in $\mathbb R^d$, then Michael Touitou's suggestion just to take a discrete metric will do the trick.