Re: Surface,which is not second countable


eugene schrieb:

eugene wrote:
It is well-known fact that any Riemann surface is second countable,
i.e. there is a countable base for its topology.

It arises the following question: Could you give an example of a
surface which is not second countable ?

Thanks.

In the sentence "It is well-known fact that any Riemann surface is
second countable" i meant

It is well-known fact that any CONNECTED Riemann surface is second
countable,

A Riemann surface is /by definition/ a manifold and a manifold is /by
definition/ second countable.
Still, the simplest "counterexamples" are not connected:
Let S be any discrete topological space, then SxC "looks like" a
Riemann surface - but it fails to
be second countable if S is not countable.

For another famous topological space that is almost a manifold except
that it fails to be second countable, check out the "long line"
http://en.wikipedia.org/wiki/Long_line_%28topology%29 (from which one
can produce a "wide plane"(?))

.

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