How to Show $(0,\infty ) $ is connected in $R_k$ topology?

Question

I wanted to show that $(0,\infty ) $ is connected in $R_k$ topology.

OPen set in $R_k$ is $(a,b)$ or $(a,b)\setminus K $where $K=\{1/n \mid n\in \mathbb N\}$

As open interval In $(1,\infty ) $ in R and k topology same so connected On the contrary if not

$(0,1) = C\cup B$ where $C$ and $B$ are open

How to arrive at the contradiction that I don't get

Please, can anyone give me a hint so that I can complete this?

ANy Help will be appreciated

Answer 1

On $(0, \infty)$ the usual topology and that of $\mathbb{R}_K$ coincide, as $K$ is closed in $(0,\infty)$ in both subspace topologies. As it is connected in the usual one, it's connected in both.