Is there a topological Hausdorff space with an infinite number of isolated points such that any infinite set of isolated points have an infinite number of limit points !? (Of course it would be impossible if a limit point is a limit of a sequence.)
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Yes: $\beta\Bbb N$, the Čech-Stone compactification of $\Bbb N$, which is a compact Hausdorff space, has that property. In fact, every infinite subset of $\Bbb N$ has $2^{\mathfrak{c}}$ limit points. (Every point of $\Bbb N$ is isolated.)
Brian M. Scott
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@julien: Thanks! – Brian M. Scott Oct 26 '13 at 22:48
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Would you give me some references to study the properties of $\beta\Bbb{N}$? – user41304 Oct 27 '13 at 06:33
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@user41304: Three possibilities: Russell C. Walker, The Stone-Čech Compactification; Neil Hindman & Dona Strauss, Algebra in the Stone-Čech Compactification; Leonard Gillman & Meyer Jerison, Rings of Continuous Functions. I’m personally familiar only with the last; I’ve picked up most of what I know in bits. – Brian M. Scott Oct 27 '13 at 06:41