)\end{array}$$

with the last two being disjoint unions, and $A$ is closed iff $A$ contains all its cluster points, $\textrm{Der}(A)\subseteq A$, iff $A$ contains its closure. Hence $$\begin{gathered}
A=\textrm{Cl}(A)\Longleftrightarrow\textrm{Cl}(A)=\{ x\in A\!:((\exists N\in\mathcal{N}_{x})(N\subseteq A))\vee((\forall N\in\mathcal{N}_{x})(N\bigcap(X-A)\neq\emptyset))\}\end{gathered}$$ Comparison of Eqs. (\[Eqn: Def: Boundary\]) and (\[Eqn: Def: Derived\]) also makes it clear that $\textrm{Bdy}(A)\subseteq\textrm{Der}(A)$. The special case of $A=\textrm{Iso}(A)$ with $\textrm{Der}(A)\subseteq X-A$ is important enough to deserve a special mention:

**Definition 2.4.** ***Donor set.*** *A proper, nonempty subset $A$ of $X$ such that* $\textrm{Iso}(A)=A$ *with* $\textrm{Der}(A)\subseteq X-A$ *will be called* *self-isolated* *or* *donor.* *Thus sequences eventually in a donor set converges only in its complement; this is the opposite of the characteristic of a closed set where all converging sequences eventually in the set must necessarily converge in it. A closed-donor set with a closed neighbour has no derived or boundary sets, and will be said to be* *isolated in $X$.*$\qquad\square$

**Example 2.5.** In an isolated set sequences converge, if they have to, simultaneously in the complement (because it is donor) and in it (because it is closed). Convergent sequences in such a set can only be constant sequences. Physically, if we consider adherents to be contributions made by the dynamics of the corresponding sequences, then an isolated set is secluded from its neighbour in the sense that it neither receives any contributions from its surroundings, nor does it give away any. In this light and terminology, a closed set is a *selfish* set (recall that a set $A$ is closed in $X$ iff every convergent net of $X$ that is eventually in $A$ converges in $A$; conversely a set is open in $X$ iff the only nets that converge in $A$ are eventually in it), **whereas a set with a derived set that intersects itself and its compleme