Duality between Ideal and Filters on a poset I am studying Ideals and Filters on a poset, and it is clear to me that this are dual notions in the sense that reversing inclusions in the definition we can get the one from the other, or in the sense that taking complements of subsets belomging to a structure we obtain the associated dual one. 
I was wondering if there is more about this duality concept, particularly in relation to general topology: Filters are used directly to assign neighboroods, do Ideal directly play a role as well?

Yes there is a topological duality.
For a space S and a subset A, 
$S - A^o = \overline{S - A}$.
It is a self dual statement to $S - \overline A = (S - A)^o$.  
One is proved by the other by taking complements and interchanging A and S - A.  
An example of its usefullness:
$(A \cap B)^o = A^o \cap B^o$
is an easy theorem to prove.
The dual statement $\overline{A \cup B} = \overline A \cup \overline B$,
is harder to prove directly but is a quick result of the former by the above described process, a bit of DeMorgan juggling. and   the self dual theorem.  
To apply this to ideals, the dual notion of neighborhood is needed.
A nhood of x is any set K with x in the interior of K.
A surrounding of x is any set K with x not in the closure of K.
Ideals can can be assigned to surroundings, an unused concept.  
