Question

Is it possible to have an unbounded, closed interval? I can think of examples of pretty much anything else but this.

Answer 1

\[Bounded = [3,9) |||(-2,100)|||[0,5]|||(3,8]\]
\[Unbounded = (-\infty,5]|||(-\infty,\infty)|||(2,\infty)\]
\[Open-Bounded = (-2,100)\]
\[Open-Unbounded = (-\infty,\infty)\]
\[Closed-Bounded = [0,5]\]
\[Closed-Unbounded = ???\]

Answer 2

\[[a,+\infty)\]

Answer 3

I saw that also through ProofWiki, but could you explain? I thought that that was simply a left-bounded or half-bounded interval/left closed, right open interval. For an interval to be completely unbounded, it needs to not include its endpoints, right? Or are unbounded and half-bounded used synonymously?

Answer 4

E.g. could somebody find a left closed, left unbounded, right closed, right unbounded interval?

Answer 5

(Or do they even exist).

Answer 6

@Mertsj

Answer 7

My understanding is that an unbounded interval does not include its endpoints. And that a closed interval does include its endpoints. So how could an interval be both unbounded and closed? That like saying "give an example of a negative number that is positive."

Answer 8

Am I wrong?

Answer 9

You're right, and I thought that was the answer entirely by definition, but I couldn't find an example anywhere on the web of anybody explicitly saying that; I don't want to say i'm cowardly, but i'm very apprehensive at going forward with my own assumptions regarding things that aren't strictly outlined. Thank you.

Answer 10

yw

Answer 11

Well, I think the interval I proposed (from ProofWiki, or even Tom Apostol Mathematical Analysis) is closed unbounded because it has all its accumulation points (is closed) but can not be fully included in a closed ball (is unbounded). Take a look to the book of Apostol (for example).

Answer 12

Thanks for the reference. I have no idea what accumulation points are, nor what a closed ball is, lol, so you clearly know way more on the subject than I do, and i'll take your word for the time being while i'm catching up.