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**1. Homework Statement**

Is there a non-empty perfect set that contains no rational number?

**2. Homework Equations**

None

**3. The Attempt at a Solution**

I thought the answer was no, but my professor said that there is. My reasoning is as follows (please let me know if I'm wrong here):

If p is an irrational limit point of a perfect set P, then every open ball B(p;r) around the point such that B(p;r) that contains another point in P. But this ball contains rational numbers, so a rational number q is in B(p;r). Thus a ball of the same radius around q contains the point p, which is in P. So q is a limit point of P (because r was arbitrary). Since P must be closed, it contains all of its limit points, so q is in P.

Where'd I mess up? Thanks in advance!