In this (mostly expository) paper, we review a proof of the following old
theorem of R.L. Moore: for a closed equivalence relation on the 2-sphere such
that all equivalence classes are connected and non-separating, and not all
points are equivalent, the quotient space is homeomorphic to the 2-sphere. The
proof uses a general topological theory close to but simpler than an original
theory of Moore. The exposition is organized so that to make applications of
Moore's theory (not only Moore's theorem) in complex dynamics easier, although
no dynamical applications are mentioned here.