Here, $\mathbb{R}_l$ is the lower limit topology on $\mathbb{R}$ and $\mathbb{R}_K$ is the K-topology on $\mathbb{R}$. I understand the proof that these topologies are strictly finer than $\mathbb{R}$, but I am at a loss to begin how to show they aren't comparable. This is from Munkres book.
Show that the Topologies of $\mathbb{R}_l$ and $\mathbb{R}_K$ are not comparable.
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general-topology
1 Answers
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To show that they are not comparable, you just need to find an open set in each that is not open in the other. (As in Munkres, I will denote the set $\{ \frac{1}{n} : n \in \mathbb{Z}_+ \}$ by $K$.)
- The set $[2,3)$ is open in $\mathbb{R}_l$, but not in $\mathbb{R}_K$.
- $\mathbb{R} \setminus K$ is open in $\mathbb{R}_K$, but not in $\mathbb{R}_l$. (Every open set in $\mathbb{R}_l$ containing $0$ meets the set $K$.)
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0@Domonic: Of course you can use that characterization! (And quite often that is e$x$actly what one would do.) – 2012-09-17