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However, the reading went on to say that the boundary of the rationals Q is R. This seems wrong to me so I am questioning the entire site.

Wouldn't the boundary of Q be Q? A ball of positive radius about any point in Q would contain both points from Q as well as irrationals from P. ddQ = dQ = Q (where d is boundary operator). The theorem at top does appear to hold but the example is messed up....no?