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.
Best Answer
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.