Does an open Interval of natural numbers have a minimum and maximum? (unlike one of rational numbers)

Like for example (1, 9)

The maximum would be 8, because we are talking about natural numbers, so the problem of an undefined maximum (the number right before 9) doesn't exist?

Am I right?

• Welcome to MSE. You should choose your tags carefully. Among those tags, only the natural-numbers tag is appropriate for this question. Apr 28, 2022 at 9:11
• Any subset of the natural numbers is open using the natural definition of distance. Any bounded subset of the natural numbers has a minimum and a maximum using the natural order. Apr 28, 2022 at 9:13
• For natural numbers, $(1,9)$ is simply $[2,8]$. Apr 28, 2022 at 9:22
• @Henry: Any bounded nonempty subset. ;-) Apr 28, 2022 at 12:33

$$(1,9)$$ is by definition the set of real numbers both greater than $$1$$ and less than $$9$$. Using $$(1,9)$$ to mean something else will only cause confusion to you and others, and so there is no point in doing so unless the purpose is to cause confusion.
If you want to talk about the set of integers greater than $$1$$ and less than $$9$$, then you could write: $$\{ x:\ x\in (1,9), x\in\mathbb{N} \}\$$ or $$(1,9) \cap \mathbb{N},\$$ or $$(1,9) \cap \mathbb{Z},\$$ but the most standard way to write this set is just to write the list itself: $$\{ 2,3,4,5,6,7,8\}.$$