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I'm wondering whether the definition of closed interval given here is correct. Quoting part of it:

If one of the endpoints is $+\infty$ or $-\infty$, then the interval still contains all of its limit points (although not all of its endpoints).

For instance $[a,\infty)$ doesn't contain all of its endpoits. But wait, what is an endpoint? It's a real number, right? The interval has only one endpoint - $a$, so it does contain all of its endpoints (however it only has one).

Secondly, the definition forgot to add that $(-\infty, \infty)$ is also open.

By the way, what is the difference between an endpoint and limit point? They use the term 'limit point' in the definition of closed interval, but it doesn't appear in the definition of open interval on the same page.

user216094
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1 Answers1

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The definition is just the first sentence: "A closed interval is an interval that includes all of its limit points." The section you quote is helping to explain the definition. The article does not state that $(-\infty,\infty)$ is open because it is about closed intervals, but you are correct that it is open.

Ross Millikan
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  • Ok, but it also says that $[a,\infty)$ doesn't contain all of its endpoints. I think it does, because it has only one endpoint! – user216094 Dec 02 '15 at 16:54
  • @user216094 they specifically allow that "If one of the endpoints is $\pm\infty$" etc., so it is a matter of convention to say that the interval does not contain the endpoint $\infty$. It would have sounded better to me it they had said that the interval $[a,\infty)$ is a closed set (reserving the expression closed interval for $[a,b]$). They also link there to the definition of limit points. Calling $\infty$ an endpoint may be imprecise, but it is the usual convention (and could be made precise considering compactifications, except $[a,\infty)$ won't be a closed set then). – Mirko Dec 02 '15 at 18:50
  • @Mirko so if we treat $\infty$ as an endpoint, then $[a,\infty)$ wouldn't be a closed set (and it's not what we want, we'd like it to be a closed set, that's what everyone would answer if you asked them whether $[a,\infty)$ is closed or not). Then we shouldn't call $\infty$ an endpoint, and $[a,\infty)$ has only one endpoint. – user216094 Dec 02 '15 at 19:56
  • @user216094 you could call that type of endpoint a "gap": it is good enough to be called (informally) an endpoint, but it does not belong to the space under consideration, so $[a,\infty)$ remains closed, as the gap is actually not there to cause a problem. – Mirko Dec 02 '15 at 20:08