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

don'tneed a global bound. If all chains are finite, the poset is well founded, hence every element is above a minimal element. $\endgroup$ – Emil Jeřábek Oct 24 '14 at 10:03