## Ultrafilters Are Maximal

Ultra filter A filer $\mathcal{F}$ containing either $Y$ or $Y^\complement$ for any $Y \subseteq X$. Two days ago, I spent an afternoon to understand Dudley’s proof of this little result.
A filter is contained in some ultrafilter. A filter is an ultrafilter iff it’s maximal.
At the first glance, I didn’t even understand the organisation of the proof! I’m going to rephrase it for future reference.
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