  We show that the four-dimensional U(1) gauge theory in the continuum
formulation has a confining phase (exhibiting area law of the Wilson loop) in
the strong coupling region above a critical coupling $g_c$. This result is
obtained by taking into account topological non-trivial sectors in U(1) gauge
theory. The derivation is based on the reformulation of gauge theory as a
deformation of a topological quantum field theory and subsequent dimensional
reduction of the D-dimensional topological quantum field theory to the
(D-2)-dimensional nonlinear sigma model. The topological quantum field theory
part of the four-dimensional U(1) gauge theory is exactly equivalent to the
two-dimensional O(2) nonlinear sigma model. The confining (resp. Coulomb) phase
of U(1) gauge theory corresponds to the high (resp. low) temperature phase of
O(2) nonlinear sigma model and the critical point $g_c$ is determined by the
Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark
(charge) confinement in the strong coupling phase is caused by the vortex
condensation. Thus the continuum gauge theory has the direct correspondence to
the compact formulation of lattice gauge theory.
