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In a metric space $X$, the boundary of an open set is the set of all limit points of a discrete set.

Actually, it is in the exercise of Willard, and it states that it requires the axiom of choice.

I thought an example, $S= \bigcup_{k \in \Bbb N} \left(\frac{1}{2k}, \frac{1}{2k-1}\right)$ is an open set in $\Bbb R$ and $\operatorname{Bd}(S)= \left\{ \frac1k \mid k \in \Bbb N \right\} \cup \{0\}$, and there is a discrete set, namely $T=\left\{\frac1m+\frac1n \mid m,n \in \Bbb N \right\}$ with derived set of $T$ being $\operatorname{Bd}(S)$.

And, for any open set $U$, $\operatorname{Bd}(U)= \operatorname{cl}(U)-U$.

But how to show existence of such a set like $T$ using the known facts. And, it can be guessed that the Choice Axiom may be used in showing existence of $T$ , but how?

Henno Brandsma
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Rabi Kumar Chakraborty
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1 Answers1

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This note on Topology Atlas, by Abhijit Dasgupta, gives a complete proof. The proof uses AC in the form of Zorn's lemma. The simpler proof sketch at the end, using paracompactness of a metric space, uses AC very blatantly, and IIRC already the fact that a metric space is paracompact requires a form of choice, in that it can fail in models without AC. I'm not sure if the original statement can fail in models without choice. (My hunch: it probably can).

Henno Brandsma
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  • To Henno Brandsma, Sir, isn't there any comparatively smooth proof , may be not using AC, as Willard"s book is far old than the paper published in 2005, and it is in the exercise of it !!! – Rabi Kumar Chakraborty Jun 22 '19 at 09:11
  • "and IIRC already the fact" what is "IIRC" Sir ??? – Rabi Kumar Chakraborty Jun 22 '19 at 09:14
  • @RabiKumarChakraborty This note was an answer on the forums on topology atlas for someone doing the same exercise as you. It's not "a paper" on which Willlard based it. He came up with it, probably based on his knowledge of paracompactness, and put it in his book. I think this note/write-up is the" smoothest" one you can find. It's pretty detailed – Henno Brandsma Jun 22 '19 at 09:14
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    @RabiKumarChakraborty IIRC = old internet speech for "if I recall/remember correctly", see https://www.urbandictionary.com/define.php?term=iirc. Old usenet user... – Henno Brandsma Jun 22 '19 at 09:16
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    @RabiKumarChakraborty AC is probably needed, but that's no problem. We topologists happily greet our AC overlords! :) – Henno Brandsma Jun 22 '19 at 09:21
  • As Mary Ellen Rudin said, "Well-order everything in sight." – DanielWainfleet Jun 22 '19 at 10:14