I proved that a club $C$ in $\kappa$ has the same cardinality as $\kappa$. Is it really true ? Thanks.
Cardinality of a club
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set-theory
cardinals
1 Answers
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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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0I supposed of course that $\kappa$ is regular .... thanks. – 2012-04-16
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0@MarcMoretti: No problem. The proof of the claim "unbounded $\Rightarrow$ of order type $\kappa$" is simple. Suppose $A$ is unbounded in $\kappa$, write $A=\{a_i\mid i<\lambda\}$ and write $A_i = [a_i,a_{i+1})$ (the half-open interval) then $\bigcup A_i = \kappa$ and for all $i$ we have $|A_i|<\kappa$. Therefore $\lambda=\kappa$, as wanted. – 2012-04-16
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0ok. An other proof is this one : let $A\subset\kappa$ club. It is unbounded so its cardinality is greatest than the minimal cardinality of a cofinal subset of $\kappa$, say $cf(\kappa)$. So $cf(\kappa)\leq|A|\leq\kappa$. But $\kappa$ regular so $|A|=\kappa$ as wanted. In fact, this proof is ok for just unbounded subset of $\kappa$. Regards. – 2012-04-18
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0@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