I'm understanding this topology through doing an exercise. We say that a subset $U \subseteq \mathbb{R}^{2}$ is radially open if if for every $x \in U$ and every $u \in \Bbb R^2$ with $||u||=1$, there is an $\epsilon>0$ such $x+su \in U$ for every $s \in (- \epsilon, \epsilon)$. I already proved these sets are a topology for $\mathbb{R}^{2}$.
First question (this was not asked for the exercise) is how can I visualize this strange topology. I got a picture for my book but still don't catch it up.
Second question, if $T_{e}$ is the usual topology for $\mathbb{R}^{2}$ and $T_{r}$ is the radially topology, how do I prove that $T_{e} \subseteq T_{r}$ and $T_{r} \not \subset T_{e}$. As I dont understand the image of this topology I cannnot figure out the first contention also for the second I have only find here confusing examples.
Last question, I want to prove that for every circumference in radially plane has discrete topology as topological subspace. I think this was asked once and they always explain that this intersection gives the center of the circumference but dont really understand the steps to conclude this.
Any help will be aprecciated. Thanks!