There exist ( necessarily non-compact ) topological surfaces having no countable base for their topology.
2.
Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology.
3.
;second-countable or "'perfectly separable "'if it has a countable base for its topology.
4.
Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is "'?-algebraic " '.
5.
Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second countable space is also a first-countable space.
6.
Another surface having no countable base for its topology, but " not " requiring the Axiom of Choice to prove its existence, is the Pr�fer manifold, which can be described by simple equations that show it to be a real-analytic surface.
7.
By contrast, the existence of the Pr�fer surface shows that there exist two-dimensional complex manifolds ( which are necessarily 4-dimensional real manifolds ) with no countable base . ( This is because any " n "-real-dimensional real-analytic manifold " Q " can be extended to an " n "-complex-dimensional complex manifold " W " that contains " Q " as a real-analytic submanifold .)
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