  We consider the class ${\cal P}_1$ of all infinite words $x\in A^\omega$ over
a finite alphabet $A$ admitting a prefixal factorization, i.e., a factorization
$x= U_0 U_1U_2 \cdots $ where each $U_i$ is a non-empty prefix of $x.$ With
each $x\in {\cal P}_1$ one naturally associates a "derived" infinite word
$\delta(x)$ which may or may not admit a prefixal factorization. We are
interested in the class ${\cal P}_{\infty}$ of all words $x$ of ${\cal P}_1$
such that $\delta^n(x) \in {\cal P}_1$ for all $n\geq 1$. Our primary
motivation for studying the class ${\cal P}_{\infty}$ stems from its connection
to a coloring problem on infinite words independently posed by T. Brown in
\cite{BTC} and by the second author in \cite{LQZ}. More precisely, let ${\bf
P}$ be the class of all words $x\in A^\omega$ such that for every finite
coloring $\varphi : A^+ \rightarrow C$ there exist $c\in C$ and a factorization
$x= V_0V_1V_2\cdots $ with $\varphi(V_i)=c$ for each $i\geq 0.$ In \cite{DPZ}
we conjectured that a word $x\in {\bf P}$ if and only if $x$ is purely
periodic. In this paper we show that ${\bf P}\subseteq {\cal P}_{\infty},$ so
in other words, potential candidates to a counter-example to our conjecture are
amongst the non-periodic elements of ${\cal P}_{\infty}.$ We establish several
results on the class ${\cal P}_{\infty}$. In particular, we show that a
Sturmian word $x$ belongs to ${\cal P}_{\infty}$ if and only if $x$ is
nonsingular, i.e., no proper suffix of $x$ is a standard Sturmian word.
