i'm sorry if my question is really trivial but this one is really bugging me out..
So let's have a partially ordered set $I$ with the topology in which the open sets are the increasing ones: $i\in U$ and $j\ge i$ implies $j\in U$.
In the article i am reading the author is making this two assumptions:
1) There is a global bound in the lengths of all chains in $I$
2) for each $i$ there are only finitely many $j$ comparable to it
I can see why these two assumptions are not equivalent to one another:
$1 \nRightarrow 2$ because i could have an infinite number of chains of finite length in which $i$ appears
$2 \nRightarrow 1$ because i could have bounds which are not $global$
What i don't get is the following:
If i assume (1) then $I$ is covered by the open sets $U_{\ge i_0}=\{j|j\ge i_0\}$ for $i_0$ minimal
What i don't get is where i need the hypothesis of the $global$ bound..
condition (2) then is needed to say that $\{U_{\ge i_0}|i_0$ is minimal$\}$ is locally finite which i think is trivial