For any topological space
, we define
It has a natural structure of a ring by introducing ``point-wise operations'':
It has an extra structure of
and a topology (locally uniform topology) which we shall not describe in detail.
is a bijection.
with a certain topology.
A first interesting part of modern algebraic geometry is that
we may mimic the correspondence in the Gelfand-Naimark theorem above
and associate to any commutive ring a compact (but not Hausdorff) space
. The elements of
may then
be considered as ``continuous functions'' on
.
The upshot is that we may ``cut and paste'', as one usually does with functions, elements of abstract commutative rings. Any other method of functional analysis also has the possibility to be applied in the commutative ring theory.
On the other hand,
it is possible to manipulate the compact space
and
create new algebras out of the existing commutative ring
.
We may furtheremore paste such
's altogther and
define another geometric objects.