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Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a normal topological space.

Martin Sleziak
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user31497
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1 Answers1

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HINTS: For regularity, first prove that $\beta$ is a base for $\tau$ consisting of clopen (i.e., both closed and open) sets; regularity of $X$ follows immediately. For non-normality, let $D=\{\langle x,1-x\rangle:x\in[0,1)\}$; you can easily prove that $D$ is a closed, discrete subset of $X$. Let $H=\{\langle x,1-x\rangle\in D:x\in\Bbb Q\}$, and let $K=D\setminus H$; then $H$ and $K$ are disjoint closed subsets of $X$ that cannot be separated by disjoint open sets. For this part you’ll probably want the Baire category theorem.

Brian M. Scott
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  • I believe one could also apply Jone's Lemma to show non-normality: If $X$ contains a dense set $D$ and a closed, relatively discrete subspace $S$ with $|S|\ge 2^{|D|}$, then $X$ is not normal. c.f Willard General Topology. pg. 100. – David Mitra May 15 '12 at 21:30
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    @David: True. (For some reason I always forget about Jones’ Lemma.) – Brian M. Scott May 15 '12 at 21:37
  • Sorry, but can you please explain why X is regular. Thanks in advance – user31497 May 16 '12 at 19:59
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    @user31497: Which part are you having trouble with? Seeing that each member of $\beta$ is closed as well as open? Or seeing why that implies regularity? – Brian M. Scott May 16 '12 at 20:05
  • I think the second. – user31497 May 16 '12 at 20:18
  • @user31497: Let $x\in X$, and let $F$ be a closed set in $X$ not containing $x$. Then $X\setminus F$ is an open set containing $x$, so there is some $B\in\beta$ such that $x\in B\subseteq X\setminus F$. $B$ is closed as well as open, so $B$ and $X\setminus B$ are disjoint open sets with $x\in B$ and $F\subseteq X\setminus B$. Does that help? – Brian M. Scott May 16 '12 at 20:20
  • Yes, thank you. – user31497 May 16 '12 at 20:27
  • @Brian M. Scott Why $D$ is discrete? –  Sep 13 '18 at 05:49
  • @Brian M. Scott Dear Brian, I was also thinking about the example $H$ which is the set of rational points in the line $D={x+y=1}$, and the set $D-H$. I wasn't able to show they cannot be separated by disjoint open sets. Can you provide some details about this? For example how Baire's theorem can be applied here? Thanks! – Yuval Feb 15 '21 at 23:27
  • @Brian M. Scott Dear Brian, now I understand. Thanks for your answer (in 2012) ! – Yuval Feb 15 '21 at 23:51
  • @Yuval: You’re welcome (and I’m glad that you were able to sort it out for yourself). – Brian M. Scott Feb 15 '21 at 23:52
  • @BrianM.Scott I am very much interested in the role of Baire theorem to reject the existence of separation, thanks in advance. – Riaz Dec 09 '22 at 11:00
  • @Riaz: The details are a bit too long for a comment, but you can find them in this question and my answer to it, and in the second bullet point in this post at Dan Ma’s Topology Blog. – Brian M. Scott Dec 09 '22 at 22:06