  Since the time when the first optical instruments have been invented, an idea
that the visible image of an object under observation depends on tools of
observation became commonly assumed in physics. A way to formalize it in
mathematics is the construction that assigns to an arbitrary object $A$ in a
category $K$ its envelope $\text{Env}_\varPhi^\varOmega A$ in a given class of
morphisms (a class of representations) $\varOmega$ with respect to a given
class of morphisms (a class of observation tools) $\varPhi$. It turns out that
if we take a sufficiently wide category of topological algebras as $K$, then
each choice of the classes $\varOmega$ and $\varPhi$ defines a "projection of
functional analysis into geometry", and the standard "geometric disciplines",
like complex geometry, differential geometry and topology, become special cases
of this construction.
  This gives a formal scheme of "categorical construction of geometries" with
many interesting applications, in particular, "geometric generalizations of the
Pontryagin duality" (to the classes of non-commutative groups).
  In this paper we describe this scheme in topology and in differential
geometry.
