Visualizing weird topology Let $(\mathbb R, T)$ be a toplogical space. A set $U$ is open iff for each $p \in U$ there is an open interval $I_p$ such that $p \in I_p$ and every rational number in $I_p$ is in $U$.
The problem is to check which separation axioms hold for this space. My problem is that I can't visualize what this space looks like. I haven't been able to convince myself that anything other than the empty set and $\mathbb R$ are open. Are there any other open sets? What do the open sets look like? What about the closed sets? I would really appreciate some help visualizing this space.
 A: Let $U$ be any set that’s open in the usual topology. If $p\in U$, there is an open interval $I_p$ such that $p\in I_p\subseteq U$, so it’s certainly true that $p\in I_p$ and $I_p\cap\Bbb Q\subseteq U$. Thus, every subset of $\Bbb R$ that’s open in the usual topology is still open in $T$.
Again let $U$ be any set that’s open in the usual topology, and let $A$ be any set of irrational numbers. Let $V=U\setminus A$, and suppose that $p\in V$. Then $p\in U$, so there is an open interval $I_p$ such that $p\in I_p\subseteq U$. Finally, $A\cap\Bbb Q=\varnothing$, so $p\in I_p$, and
$$I_p\cap \Bbb Q\subseteq U\cap\Bbb Q\subseteq U\setminus A=V\;,$$
so $V\in T$.
The members of $T$ are simply the sets of the form $U\setminus A$, where $U$ is open in the usual topology, and $A$ is any set of irrational numbers. In particular, for any $x\in\Bbb R$ the sets of the form
$$\{x\}\cup\Big((x-\epsilon,x)\cap\Bbb Q\Big)\cup\Big((x,x+\epsilon)\cap\Bbb Q\Big)$$
with $\epsilon>0$ form a local base at $x$ in the space $\langle\Bbb R,T\rangle$.
