Example 2

More generally, let $f$ be a mapping of a directed set $A$ into a topological space $X$. If $x \in X$ is a limit (resp. cluster point) of $f$ with respect to the section filter of $A$ then $x$ is said to be a limit (resp. cluster point) of $f$ with respect to the directed set $A$, and we write $x = \lim_{z \in A} f(z)$.

If $y$ is a limit (resp. cluster point) of a mapping $f : X \to Y$ with respect to a filter $\mathfrak{F}$ on $X$, then $y$ remains a limit (resp. cluster point) of $f$ with respect to $\mathfrak{F}$ if we replace the topology of $Y$ by a coarser topology, or if we replace the filter $\mathfrak{F}$ by a finer (resp. coarser) filter.