1. ## K-topology on R

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 $\displaystyle K\cup (a,b)-K$.

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?

2. Originally Posted by 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?
Lemma 1. If Y is a connected subspace of a space X, then $\displaystyle \bar{Y}$ 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 $\displaystyle \mathbb{Re}$ are intervals.

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

,
,
,

,

,

,

,

,

,

,

,

,

# show that K topology is connected

Click on a term to search for related topics.