4
$\begingroup$

We know that if $X$ is a second countable and regular, then it is normal. Now assume that $X$ is second countable and Hausdorff. Are there any examples to show that $X$ need not necessarily be a normal space?

$\endgroup$
3
4
$\begingroup$

Consider the subset $X=\mathbb{Q}\times \{0\}\cup \mathbb{R}\times\mathbb{R}_{>0}$ of $\mathbb{R}\times \mathbb{R}$. For each $q\in \mathbb{Q}$ and $r\in \mathbb{Q}_{>0}$, we set $$ D_{q,r}=\{(q,0)\}\cup \{(x,y)\in \mathbb{R}\times \mathbb{R}_{>0}\mid (x-q)^2+y^2<r\}\subset X. $$ Equip $X$ with the weakest topology such that

  • it is stronger than the topology induced from $\mathbb{R}\times \mathbb{R}$, and
  • $D_{q,r}$ is open for each $q\in \mathbb{Q}$ and $r\in \mathbb{Q}_{>0}$.

Then $X$ is second countable and Hausdorff, but we cannot separate $(0,0)$ and the closed subset $(\mathbb{Q}\setminus\{0\})\times \{0\}$ by open neighborhoods.

$\endgroup$
3
$\begingroup$

Maybe the simplest example is Munkres' favourite example $\Bbb R_K$, see here, e.g.

It's Hausdorff, non-regular, separable and second countable.

Tip: to look for such examples in the future you can use $\pi$-base queries. Very useful, and highly recommended; the previous example is called Smirnov's deleted sequence topology there.

$\endgroup$
1
  • 1
    $\begingroup$ Wow! This is a great database! Thank you so much for sharing $\endgroup$ Sep 26 at 22:32

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.