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For context, let me clarify some things. Our set is $\mathbb{R}$. The topology on $\mathbb{R}$ is the topology generated by the arbitrary unions of closed-open intervals $[a,b)$ with $a,b \in \mathbb{R}$. I've been told that this is a classic example of a topological space that is separable but not second countable.

What is it exactly that makes this space not second countable?

Also, just so I can garner a better understanding, how could we change $[a,b)$ so that it becomes second countable? Is it the closed bracket that's causing the problem or the open bracket? It's hard to tell. I think it's the closed bracket because if we had two open brackets, that would just be the natural topology on $\mathbb{R}$ which is second countable. So it must be the closed bracket that throws us off.

I'm just not understanding which part of the lower limit topology violates the definition of second countable.

Jose Avilez
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SunRoad2
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  • Does this help? https://math.stackexchange.com/questions/3983378/why-is-the-sorgenfrey-line-not-second-countable – Jose Avilez May 30 '21 at 03:40
  • I just looked at your link. It helps me understand why $a,b \in \mathbb{Q}$ wouldn't produce the Sorgenfrey line, but it fails to help me understand the countability part. I think part of my problem is understanding how $(a,b)$ with $a,b \in \mathbb{R}$ yields a second countable space. – SunRoad2 May 30 '21 at 03:44
  • Try to distinguish between "countable" and "first/second countable". The space $\mathbb{R}$ with the usual topology is uncountable, first countable, and second countable. The Sorgenfrey line is uncountable, first countable, and not second countable. While I've not studied topology past a "quals" level, it sounds wrong to me e.g. to say "uncountable" to mean "not second countable"... – Brian Moehring May 30 '21 at 04:04
  • @BrianMoehring you're right. I think I'm confusing myself between the conventional notion of countability and the notions of first and second countability – SunRoad2 May 30 '21 at 04:06

1 Answers1

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Start by recalling why $\mathbb{R}$ with the usual topology is second-countable. We take (for example) the set of open intervals with rational endpoints. Now something like $(\pi, 10)$ isn't in this set, but it can be built from elements of this set as an appropriate union: fixing a descending sequence $10>a_1>a_2>...$ with $\lim_{i\rightarrow\infty}a_i=\pi$, we have $$(\pi,10)=(a_1, 10)\cup (a_2,10)\cup (a_3,10)\cup ...$$

The crucial point here is that the "dangerous point" $\pi$ does not have to be in the set $(\pi,10)$ which we're trying to build, so approaching it from the right is good enough for us. By contrast, when we look at the Sorgenfrey topology we have $$[\pi,10)\color{red}{\not=}[a_1, 10)\cup [a_2,10)\cup [a_3,10)\cup ...$$ Instead, the right hand side is just $(\pi,10)$. So the argument that the usual topology on $\mathbb{R}$ is second-countable breaks down for the Sorgenfrey topology - the culprit being exactly the "square bracket" part.

Noah Schweber
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