A Berkovich space is a non-Archimedean analytic space whose local charts include multiplicative seminorms as points, rather than only
points with coordinates in finite extensions
of the ground field. For an algebra of convergent power
series over a complete non-Archimedean field 
, modulo an ideal, its Berkovich spectrum consists of
bounded multiplicative seminorms extending the absolute
value on 
. The topology is the weakest one making evaluation
on each algebra element continuous (Conrad 2008).
For example, the Berkovich affine line consists of multiplicative seminorms on ![K[T]](/images/equations/BerkovichSpace/Inline3.svg)
extending that absolute value. Besides evaluation
at field-valued points, it contains disk seminorms
such as
 |
where 
and 
. These extra points allow analytic disks and their
attachment points to encode connected geometry.
-adic
Geometry: Lectures from the 2007 Arizona Winter School (Ed. D. Savitt and
D. S. Thakur). Providence, RI: Amer. Math. Soc., pp. 9-63, 2008.