1
$\begingroup$

I have been goind through Rudin for the past month and I have arrived at this theorem in the topology chapter. Now I understand Rudin's proof but I was trying to come up with my own and I am not sure if it is correct or not.

Theorem: Every neighbourhood is an open set.

Proof: Suppose that there exists a neighbourhood $N$ of $p$ that is not open. Then there exists a point $q$ in $N$ that is not an interior point of $N$. Then any neighbourhood $N_q$ of $q$ is not included in $N$. Choose $N_q$ such that $p$ is in $N_q$. Since $N_q$ is not in $N$, then $p$ is not in in $N$. Contradiction.

I have attached an image showing what I had in mind when I wrote this.

The gray edge boundary means that it's not included

$\endgroup$
  • $\begingroup$ What is your definition of a neighbourhood? $\endgroup$ – user2520938 Sep 10 '17 at 19:16
  • $\begingroup$ A neighbourhood of a point p is a set N consisting of all points q such that d(p, q) < r, where r>0. $\endgroup$ – user3535525 Sep 10 '17 at 19:21
  • $\begingroup$ Oke, then what is your definition of an open set? $\endgroup$ – user2520938 Sep 10 '17 at 19:21
  • $\begingroup$ A set is open if every point of that set is an interior point of that set. And a point p is interior of E if there is a neighbourhood N of p such that N is in E. $\endgroup$ – user3535525 Sep 10 '17 at 19:23
  • $\begingroup$ You should probably include this in the question, since they are equivalent but non-standard definitions. $\endgroup$ – user2520938 Sep 10 '17 at 19:25
0
$\begingroup$

I think your proof is incorrect. There is a problem with your statement: "Since $N_q$ is not in $N$, then $p$ is not in $N$".

It is true that $N_q$ is not a subset of $N$, but it is certainly possible that some elements (points) of $N_q$ are also elements of $N$.

$\endgroup$
  • $\begingroup$ You are right, I see my error now. Thank you. $\endgroup$ – user3535525 Sep 10 '17 at 19:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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