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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.

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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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    $\begingroup$ Thanks for your response! When determining whether or not the set is open, can I go by the criterion used in section 13, namely, $U$ is open if for each element in U, there is a basis element $B$ such that x is an element of $B$ and $B$ is a subset of $U$? $\endgroup$ – madisonfly Sep 16 '12 at 22:14
  • $\begingroup$ @Domonic: Of course you can use that characterization! (And quite often that is exactly what one would do.) $\endgroup$ – user642796 Sep 17 '12 at 2:02

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