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I proved that a club $C$ in $\kappa$ has the same cardinality as $\kappa$. Is it really true ? Thanks.

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Every club is an unbounded set. If $\kappa$ is regular this means that every unbounded set has order type $\kappa$ and therefore of size $\kappa$.

If $\kappa$ is singular then it is not true, take \{\aleph_\alpha\mid\alpha<\omega_1\} as a club of $\aleph_{\omega_1}$.

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    @Marc: Your proof is good, better than mine. As you noticed, indeed the proof only required unboundedness and not closure of any kind.2012-04-18