  We consider the Zariski space of all places of an algebraic function field
$F|K$ of arbitrary characteristic and investigate its structure by means of its
patch topology. We show that certain sets of places with nice properties (e.g.,
prime divisors, places of maximal rank, zero-dimensional discrete places) lie
dense in this topology. Further, we give several equivalent characterizations
of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding
problems over large fields}. We also study the question whether a field $K$ is
existentially closed in an extension field $L$ if $L$ admits a $K$-rational
place. In the appendix, we prove the fact that the Zariski space with the
Zariski topology is quasi-compact and that it is a spectral space.
