of.** Follows immediately from the definitions for the closure of a set Eq. (\[Eqn: Def: Closure\]) and the adherence of a filter Eq. (\[Eqn: adh filter\]). As always, it is a matter of convenience in using the basic filters **$_{\textrm{F}}\mathcal{B}$** instead of $\mathcal{F}$ to generate the adherence set.$\qquad\blacksquare$

It is infact true that the limit sets $\lim(\mathcal{F})$ and $\lim(\chi)$ are also closed set of $X$; the arguments involving ultrafilters are omitted.

Similar to the notion of the adherence set of a filter is its *core —* a concept that unlike the adherence, is purely set-theoretic being the infimum of the filter and is not linked with any topological structure of the underlying (infinite) set $X$ — defined as $${\displaystyle \textrm{core }(\mathcal{F})=\bigcap_{F\in\mathcal{F}}F.}\label{Eqn: core}$$ From Theorem A1.3 and the fact that the closure of a set $A$ is the smallest closed set that contains $A$, see Eq. (\[Eqn: closure\]) at the end of Tutorial4, it is clear that in terms of filters$$\begin{aligned}
A & = & \textrm{core}(\,_{\textrm{F}}\mathcal{P}(A))\nonumber \\
\textrm{Cl}(A) & = & \textrm{adh}(\,_{\textrm{F}}\mathcal{P}(A))\label{Eqn: PrinFil_Cl(A)}\\
 & = & \textrm{core}(\textrm{Cl}(\,_{\textrm{F}}\mathcal{P}(A)))\nonumber \end{aligned}$$ where $_{\textrm{F}}\mathcal{P}(A)$ is the principal filter at $A$; thus *the core and adherence sets of the principal filter at $A$ are equal respectively to $A$ and* $\textrm{Cl}(A)$ *—* a classic example of equality in the general relation $\textrm{Cl}(\bigcap A_{\alpha})\subseteq\bigcap\textrm{Cl}(A_{\alpha})$ — but both are empty, for example, in the case of an infinitely decreasing family of rationals centered at any irrational (leading to a principal filter-base of rationals at the chosen irrational). This is an important example demonstrating that *the infinite intersection of a non-empty family of (closed) sets with the finite intersection property may be empty,* *a situation that cannot arise on a finite set or an in