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Consider the assertion:

Every connected, but not necessarily paracompact, n-manifold is of cardinality $2^{\aleph_0}$ (at least assuming the axiom of choice).

For n=1 this may be proved via enumeration of the short list of exampelsexamples. The essential point is that while there is a Long Line, there is no Extra Long Line.

What is the situation for n>1?

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Consider the assertionnassertion:

Every connected, but not necessarily paracompact, n-manifold is of cardinality $2^{\aleph_0}$ (at least assuming the axiom of choice).

For n=1 this may be proved via enumeration of the short list of exampels. The essential point is that while there is a Long Line, there is no Extra Long Line.

What is the situation for n>1?

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Consider the assertionn:

Every connected, but not necessarily paracompact, n-manifold is of cardinality 2^\Aleph_0 $2^{\aleph_0}$ (at least assuming the axiom of choice).

For n=1 this may be proved via enumeration of the short list of exampels. The essential point is that while there is a Long Line, there is no Extra Long Line.

What is the situattion situation for n>1?

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