Why are we interested in the K topology?

Question

I'm curious what is the importance of the K-topology on $\mathbb{R}$ as I don't see it being used as a counterexample to many problems in topology (so far) that much. I would like to ask what we can use the K-topology for that other topologies on $\mathbb{R}$ can't do. I have not done algebraic topology yet, but so far in Munkres (part 1) I don't see the K-topology being invoked that many times. (I think the only one I remember is that it's Hausdorff but not regular)

Answer 1

It's not important by itself, but a simply constructed example (so suitable for a text book) of a space $X$ that is Hausdorff but not regular, but also second countable. So it shows that in the theorem that a second countable regular space is metrisable (Urysohn) we cannot weaken regular to Hausdorff, e.g. Also: a Lindelöf regular space is paracompact and thus normal (and this example shows again we cannot weaken regular to Hausdorff).

It's a didactical example to illustrate the bounds on some theorems that Munkres treats.