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**putnam120** Let $\displaystyle K=\left\{\frac{1}{n} : n\in\mathbb{N}\right\}$. Define $\displaystyle \mathbb{R}_K$ to be the topology on $\displaystyle \mathbb{R}$ generated by the topology $\displaystyle K\cup (a,b)-K (typo?)$.

I have already shown that [0,1] is not compact in this topology. I just need to show that $\displaystyle \mathbb{R}_K$ is connected. Obviously the sets $\displaystyle (-\infty,0)\cup (1,\infty)$ are connected. All I need is to show that the range inbetween is also connected. Where should I start?