Definition: Ultrafilter
If $F$ is a filter on a set $X$ and there is no filter $G$ on $X$ that is strictly finer than $F$, then $F$ is called an ultrafilter on $X.$
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References
Bibliography
 Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
 Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition
 Grotemeyer, K.P.: "Topologie", B.I.Wissenschaftsverlag, 1969