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For example, is $\{0\}$ considered a closed interval? Why or why not? Doesn't it contain all (it's only) limit point of $0$?

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    $\begingroup$ "singular point" means something very different. The term for a one-element set is "singleton". $\endgroup$
    – BrianO
    Commented Dec 8, 2015 at 1:21
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    $\begingroup$ That last question proves that it's closed, so the only question left is, "Is it an interval"? (Yes; $\{0\}=[0,0]$.) $\endgroup$ Commented Dec 8, 2015 at 1:22
  • $\begingroup$ New question: Is $\varnothing$ considered to be an interval? I'm going to guess that even though $\varnothing=[1,0]$, people make a special exception for it (like $1$ and primes). Or perhaps it depends on who you ask. $\endgroup$ Commented Dec 8, 2015 at 1:25

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Intervals are by definition connected subsets of $\Bbb{R}$. Singletons are connected and closed. Therefore they qualify as closed intervals.

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    $\begingroup$ Unfortunately it is very often the case that in some fields the term "interval" means a connected subset of $\mathbb R$ that contains more than one point (also frequently assumed to be bounded). For example, most any research paper having to do with integration theory (Denjoy, Perron, Henstock, Burkill, Cesaro, etc. integrals). $\endgroup$ Commented Dec 8, 2015 at 16:25
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    $\begingroup$ Thank you for that comment, I didn't know that. This shows a recurrent problem in mathematics, we can't even agree on the definition of simple notions such as intervals! $\endgroup$
    – Nitrogen
    Commented Dec 8, 2015 at 16:37
  • $\begingroup$ (-1) The elaborate notion of connectedness is irrelevant here. "By definition" (like in any ordered set) $I\subset\Bbb R$ is an interval iff $\forall a,b\in I,\forall x\in\Bbb R (a<x<b\implies x\in I).$ (I wonder why @JoséCarlosSantos was downvoted today and immediately deleted his recent answer.) And closedness of singletons must be proved. $\endgroup$ Commented Mar 25, 2023 at 11:51

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