  In this article we give a homological characterization of the topology of
Stein spaces over any valued base field. In particular, when working over the
field of complex numbers, we obtain a characterization of the usual Euclidean
(transcendental) topology of complex analytic spaces. For non-Archimedean base
fields the topology we characterize coincides with the topology of the
Berkovich analytic space associated to a non-Archimedean Stein algebra. Because
the characterization we used is borrowed from a definition in derived geometry,
this work should be read as a contribution towards the foundations of derived
analytic geometry.
