Question

Construct a bounded set of real numbers with exactly three limit points?
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Answer 1

Answer:

We are to construct a bounded set of real numbers with exactly three limit points. Seeing as there are few "computation"-tasks in the book, I haven't really had a chance to build up intuition the way I am used to, however, loeoking at the text in Topology, it seems like I have to get used to this.

I let X be the metric space R. I let E be its subset {(1−ϵ,1+ϵ),(2−ϵ,2+ϵ),(3−ϵ,3+ϵ)} for small ϵ>0. Clearly, 1,2 and 3 are limit points of E. However, if I have understood the concept of a limit point (and a neighborhood) correcly, so is 1+ϵ2. I'm having a hard time seeing how I am to limit the number of limit points (no pun intended.)