I am trying to prove that the real line and the Sorgenfrey line have the same dense subsets. That is, $A\subset \Bbb R$ is dense under the lower limit topology if and only if it is dense under the usual topology.
I know that a dense set is defined as:
Let $(X, \tau )$ be a topological. We say that $D \subset X$ is dense in $X$ se $\overline{D} = X$.
My idea would be to maybe show that: (1) any set A dense in the real line is also dense in the Sorgenfrey line, (2) any set A dense in the Sorgenfrey line is also dense in the real line. But I'm not sure how to do this. Any help?
Let us denote real line by $\mathbb{R}$ and Sorgenfrey line by $\mathbb{R'}$. Then we have to prove that an arbitrary dense set $\mathbb{A}$ is dense in $\mathbb{R'}$ iff it is dense in $\mathbb{R}$.
Step 1) Assume A is dense in $\mathbb{R'}$. We will prove A is dense in $\mathbb{R}$. Let $x \in \mathbb{R}$ and $(x-\epsilon,x+\epsilon)$ be an arbitrary open set. Then $[x-\epsilon/2,x+\epsilon/2)$ contains an element 'a' from $\mathbb{A}$. This implies $(x-\epsilon,x+\epsilon)$ contains 'a'. This proves our case.
Step 2) The other way round follows a very similar argument.