Remark
58
[237]Suppose that the set \(J\) is directed but not filtering; then by [06V] it admits a maximum element that we call \(\infty \); the above definitions and properties can also be stated in this case, but they are trivial, since
\[ \lim _{j\in J} f(j) = \liminf _{j\in J} f(j) = \limsup _{j\in J} f(j) = f(\infty )\quad . \]