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.


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 $K$.)
  • 2
    $\begingroup$ 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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