I find the following to be amazing: On one hand, every Riemann surface -- i.e., 1-dimensional complex analytic manifold -- has a countable base for its topology. (Due to Tibor Rado.) On the other hand, there exist real analytic surfaces that have an uncountable base for their topology. (The standard example is the Pruefer manifold, an astonishingly simple construction that is almost always described in the opaquest possible terms. First described in print by Rado.) But given such a real analytic surface S with an uncountable base for its topology, it can always be extended to a 2-dimensional complex analytic manifold S* which must then have an uncountable base for its topology. (Of course as a real manifold S is 4-dimensional.) CONCLUSION: Complex analytic manifolds must have a countable base if they are 1-dimensional, but can have an uncountable base if their dimension is >= 2. (Of course all manifolds mentioned are assumed to be connected.) --Dan Sometimes the brain has a mind of its own.