Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

It seems to me that the basis for the K- Topology and the basis for the standard topology generate the same open sets. For instance, the open sets in the K-topology's basis that are different from the standard topology's basis are the sets which are of the form $(a,b)\backslash K $. This seems to be just the union of a countable number of disjoint intervals (which we can generate from the basis elements of the standard topology). To be more explicit, suppose we have a set in the K topology $(0,1)\backslash K$. We can generate this open set with $\cup_{n\in\mathbb{Z+}} (\frac{1}{n+1},\frac{1}{n})$, where each of the open sets $(\frac{1}{n+1},\frac{1}{n})$ is in the basis for the standard topology. Also, since the basis elements of the standard topology are a subset of the K-topology's basis elements, any open set the standard topology generates can also be generated by the K-topology. I don't see how one is strictly finer than the other but according to Munkres, the K-topology is strictly finer than the standard topology.

share|cite|improve this question
Just a question: What does K-Topology mean? I've never heart of it. – Stefan Hamcke Jan 15 '13 at 12:01
K- topology is the topology on R generated by the basis that consists of the open intervals $(a,b)$, as well as the sets $(a,b)-K$, where $K=\{\frac{1}{n}\, n\in \mathbb{N}\}$ – Akt904 Jan 23 '13 at 5:16

Let $U=\Bbb R\setminus K$. By definition $U$ is open in the $K$-topology on $\Bbb R$, but it is not open in the usual topology: $0\in U$, but $U$ does not contain any open interval around $0$.

share|cite|improve this answer
NWU Topology thanks you. :) – Austin Mohr Jan 28 at 21:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.