1. ## K-topology on R

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

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

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

I have already shown that [0,1] is not compact in this topology. I just need to show that $\mathbb{R}_K$ is connected. Obviously the sets $(-\infty,0)\cup (1,\infty)$ are connected. All I need is to show that the range inbetween is also connected. Where should I start?
Lemma 1. If Y is a connected subspace of a space X, then $\bar{Y}$ is connected.
Lemma 2. The union of a collection of connected subspace of X that have a point in common is connected.
Lemma 3. The connected subsets of $\mathbb{Re}$ are intervals.

We already know that $(-\infty,0)\cup (1,\infty)$ is connected with respect to K-topology on $\mathbb{Re}$, which denoted as $\mathbb{Re}_K$. By lemma 1 & 2, we only remain to show that (0,1) is connected with respect to $\mathbb{Re}_K$. In standard topology on $\mathbb{Re}$, (0,1) is connected by Lemma 3. Since $\mathbb{Re}_K$ is finer than the standard topology on $\mathbb{Re}$, we need to check added open sets in (0,1) by $\mathbb{Re}_K$. Since a basis element has the form $(a,b)-K$, added open sets are simply unions of intervals. Now the whole (0,1) can be described as a union of connected intervals having intersections. Thus (0,1) is connected in $\mathbb{Re}_K$ by lemma 2 & 3.

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# show that K topology is connected

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