  Let $\left\{ \mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be an infinite
sequence of families of compact connected sets in $\mathbb{R}^{d}$. An infinite
sequence of compact connected sets $\left\{ B_{n} \right\}_{n\in \mathbb{N}}$
is called heterochromatic sequence from $\left\{ \mathcal{F}_{n}\right\}_{n \in
\mathbb{N}}$ if there exists an infinite sequence $\left\{ i_{n} \right\}_{n\in
\mathbb{N}}$ of natural numbers satisfying the following two properties: (a)
$\{i_{n}\}_{n\in \mathbb{N}}$ is a monotonically increasing sequence, and (b)
for all $n \in \mathbb{N}$, we have $B_{n} \in \mathcal{F}_{i_n}$. We show that
if every heterochromatic sequence from $\left\{ \mathcal{F}_{n}\right\}_{n \in
\mathbb{N}}$ contains $d+1$ sets that can be pierced by a single hyperplane
then there exists a finite collection $\mathcal{H}$ of hyperplanes from
$\mathbb{R}^{d}$ that pierces all but finitely many families from $\left\{
\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$. As a direct consequence of our
result, we get that if every countable subcollection from an infinite family
$\mathcal{F}$ of compact connected sets in $\mathbb{R}^{d}$ contains $d+1$ sets
that can be pierced by a single hyperplane then $\mathcal{F}$ can be pierced by
finitely many hyperplanes. To establish the optimality of our result we show
that, for all $d \in \mathbb{N}$, there exists an infinite sequence $\left\{
\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ of families of compact connected
sets satisfying the following two conditions: (1) for all $n \in \mathbb{N}$,
$\mathcal{F}_{n}$ is not pierceable by finitely many hyperplanes, and (2) for
any $m \in \mathbb{N}$ and every sequence $\left\{B_n\right\}_{n=m}^{\infty}$
of compact connected sets in $\mathbb{R}^d$, where $B_i\in\mathcal{F}_i$ for
all $i \geq m$, there exists a hyperplane in $\mathbb{R}^d$ that pierces at
least $d+1$ sets in the sequence.
