At the beginning I think that I should warn everybody reading this post: I don't know much about algebraic geometry so most of specialists in this subject may seen my question as ignorant. To be honest, I made several attempts to get convinced about algebraic geometry but unfortunately, still I'm not convinced. As far I understood one on the main themes in algebraic geometry is to pursue as far as it is possible the duality between geometric objects and algebras: most basic result is the Hilbert Nullstellensatz but the theory goes much further-to the definition of general schemes due to Grothendieck. The notion of space has evolved through the history of mathematics but as far as some topological space was around, the minimal requirement (at least for me) was that the space should be Hausdorff. This is quite natural due to the following characterisation: each net has at most one limit. Moreover, when one one is interested in compact or locally compact spaces, the assumpion of being Hausdorff automatically implies better behaviour (normality or complete regularity resp.). Finally, there is the theory (which is close to my heart) of $C^*$-algebras: in this theory the fundamental result is the Gelfand-Najmark theorem which establishes the duality between compact Hausdorff spaces and commutative unital $C^*$-algebras which is another algebra-geometry duality. This result alows one to think of the theory of general $C^*$-algebras as the noncommutative topology: but there are plenty of situations when one have a "patological" topological space (with some non Hausdorff topology) which is hard to deal with. Then one switches to the realm of algebras and tries to say something about this space using the associated algebra: in other words one doesn't stick to geometric picture. It seems that algebraic geometry goes the other way around and works very often with topological spaces which are non Hausdorff. So my (rather vaque) question is the following:
Question What is the meaning and the intuition behind non Hausdorff spaces in the realm of algebriac geometry? How to interpret such non Hausdorff topologies in this algebra-geometric context?