# Derived Set of a given subset of Real Line.

Let $A = \{a +\pi b : a , b \in \mathbb{Z}\}$ is a subset of $\mathbb{R}$. What will be the derived set of it?

-
Derived set in what sense? –  Asaf Karagila Jul 15 '12 at 17:06
I think Varun wants the set of accumulation points of $A$. –  Siminore Jul 15 '12 at 17:12
Yes, Sir By Derived set, I mean the set of all limit points of the set $A$. –  shane Jul 15 '12 at 17:50

$A$ is an additive subgroup of $\Bbb R$, which is not discrete since $\pi$ is irrational. A known result about additive sub-groups of $\Bbb R$ shows that $A$ is dense in $\Bbb R$ for the usual topology. Each point which is not in $A$ is a limit point.
For the points which are in $A$, we use irrationality criteria (proposition 4 in the link).