Corollary 3

Let $A$ be a set of the filter $\mathfrak{G}$, let $\mathfrak{G}_A$ be the filter induced on $A$ by $\mathfrak{G}$, and let $f_A$ be the restriction of $f$ to $A$; then

For every cluster point of the filter base $f(\mathfrak{G})$ is a cluster point of the filter base $f_A(\mathfrak{G}_A)$, and conversely.

For this reason, if $f$ is defined only on a subset $A$ of $X$ belonging to $\mathfrak{G}$, we shall often write $\limsup_{\mathfrak{G}} f$ instead of $\limsup_{\mathfrak{G}_A} f_A$, by abuse of language.