What is the difference between $\mathbb{R}$ with the $K$ topology and $\mathbb{R}_k$?

Question

I am studying the $K$ topology and struggling with some of the basics...

1. My first question is the question in my title. What is the difference between $\mathbb{R}$ with the $K$ topology and $\mathbb{R}_K$? I think they are one and the same...correct? That means it is a collection of all open intervals along with all open intervals with unit fractions removed.

2. My second question is about the definition of the $K$ topology (although it is not really a question). It seems strange to me - just append a bunch of additional sets to the standard topology on $\mathbb{R}$. I suppose there is not much to understand about a definition...it just it, but it seems unnatural to me compared to other topologies we have studied (lower limit, order). Are there any comments to help me understand this?

3. I know $K$ is not closed in $\mathbb{R}$ because it does not contain its limit point $0$. Is $K$ closed relative to the $K$ topology because its complement is open (open intervals)?

I can prove the $K$ topology is strictly finer than the standard topology but will also want to discuss Hausdorff, connected (possibly path-connected), and compact. I am hoping some basic clarification of the above topics will assist me with this.

Thank you.

Answer 1

1. Yes. $\mathbb{R}_K$ is just notation for the topological space $(\mathbb{R}, \tau_K)$, where $\tau_K$ is the $K$-topology with basis $\{(a, b): a, b\in \mathbb{R}\}\cup\{(a, b)\setminus K : a, b\in \mathbb{R}\}$.
2. Its purpose is to be weird and unnatural: to have odd properties, so it can serve as a counter-example to some intuitive but false conjectures one might propose. It should seem unnatural because it is unnatural.
3. Yes. $K=\{\frac{1}{n} : n \mbox{ is a natural number } >0\}$ is not closed in the standard topology on $\mathbb{R}$, but is closed in the $K$-topology, for the reasons you stated.