I am unable to think how to prove this question.
Question is - Let A be a subgroup of Real Line under Addition. Show that either A is dense in Real Line Or else the subspace topology of A is discrete topology.
I tried by assuming A is not dense in Real Line but I am unable to prove that A intersection is discrete topology.
Any help will be really appreciated.
Hint: Either $A$ has a least positive element or it doesn't. The former gives you a discrete set and the latter a dense set (provided $A \neq \{0\}$).
EDIT: Adding details for the case where $A$ has a least positive element.
Let $\epsilon \in A$ be this least positive element.
Claim. $|x - y| > \epsilon/2$ for all $x, y \in A$ with $x\neq y$.
Proof. Suppose not. Let $x, y \in A$ be distinct such that $|x - y| \le \epsilon/2$.
Set $d = x - y$. Note that $d \in A$ and $-d \in A$. Thus, we may assume that $d > 0$.
Thus, we have $0 < d \le \epsilon/2 < \epsilon$. This contradicts that $\epsilon$ is the smallest positive element of $A$.
Thus, the claim is proven and it follows that $A$ is discrete.
