= Stationarity of ordinals of prescribed cofinality
{title2=$S_\theta^\kappa=\{\delta<\kappa:\operatorname{cf}(\delta)=\theta\}$}
For a regular infinite $\theta<\kappa$, with $\kappa$ a regular uncountable <cardinal number>, $S_\theta^\kappa=\{\delta<\kappa:\operatorname{cf}(\delta)=\theta\}$ is stationary. Build a strictly increasing continuous $\theta$-sequence in any <club set>, and take its supremum. Closure places it in that club and the <cofinality of an increasing ordinal supremum> gives <cofinality> $\theta$. In particular the two disjoint sets $S_\omega^{\omega_2}$ and $S_{\omega_1}^{\omega_2}$ show that the <club filter> on $\omega_2$ is not an <ultrafilter>.
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