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In the book I am reading now, Calculus, a Complete Course b Adams and Essex, in the section on continuous functions there is the term "a closed, finite interval". The book defines "a closed interval" as to be consisting of all real numbers $x$ satisfying $a \leq x \leq b$. I cannot see how a closed interval might not be finite. I read some threads and it seems to me that the concept of "finite", when one discusses sets, is different and it means that when a set is said to be finite it means it has a finite number of elements. But here $a \leq x \leq b$ we have infinite number of elements. This confuses me.

Kaveh Rad
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    I think finite means bounded interval i.e end points are finite. For an example $[0, \infty) $ is a closed interval but not bounded. – Sourav Ghosh Aug 25 '22 at 13:47
  • @SouravGhosh Thank you for the comment. This is getting more complicated. I thought $[0, \infty)$ is half-open. – Kaveh Rad Aug 25 '22 at 13:51
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    ... and $[0,\infty)$ is closed because (many equivalent reasons) a sequence inside $[0,\infty)$ must, if it converges at all, converge within $[0,\infty)$. So, it is “closed” because the sequence cannot “escape” :) whereas $(0,\infty)$ is not closed (e.g. the sequence $1/n$ “escapes”) and in fact this interval is “open” – FShrike Aug 25 '22 at 13:52
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    Kaveh, indeed it looks half open. But it is actually “closed” at the $\infty$ end (this is very handwavy, really it’s better to define “closed” using topology!) because there’s nowhere left to go. $\infty)$ is the end of the line, so to speak – FShrike Aug 25 '22 at 13:54
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    That book uses finite in finite interval in reference to its length, rather than its cardinality (number of elements). They wanted to avoid using words like compact or bounded thinking that this is better for the students to grasp, and create confusion like this. – plop Aug 25 '22 at 14:07
  • @user85667 Thank you so much. Now, I get it and it makes perfect sense considering the text. – Kaveh Rad Aug 25 '22 at 14:38

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Consider the interval (set) $[0,1] \subseteq \mathbb{R}$. This interval is closed and not finite because there are clearly infinitely many rationals and therefore infinitely many reals in this set.

Kookie
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