  A Peano continuum means a locally connected continuum. A compact metric space
is called a \emph{Peano compactum} if all its components are Peano continua and
if for any constant $C>0$ all but finitely many of its components are of
diameter less than $C$. Given a compact set $K\subset\mathbb{C}$, there usually
exist several upper semi-continuous decompositions of $K$ into subcontinua such
that the quotient space, equipped with the quotient topology, is a Peano
compactum. We prove that one of these decompositions is finer than all the
others and call it the \emph{core decomposition of $K$ with Peano quotient}.
This core decomposition gives rise to a metrizable quotient space, called the
Peano model of $K$, which is shown to be determined by the topology of $K$ and
hence independent of the embedding of $K$ into $\mathbb{C}$. We also construct
a concrete continuum $K\subset\mathbb{R}^3$ such that the core decomposition of
$K$ with Peano quotient does not exist. For specific choices of
$K\subset\mathbb{C}$, the above mentioned core decomposition coincides with two
models obtained recently, namely the locally connected model for unshielded
planar continua (like connected Julia sets of polynomials) and the finitely
Suslinian model for unshielded planar compact sets (like polynomial Julia sets
that may not be connected). The study of such a core decomposition provides
partial answers to several questions posed by Curry in 2010. These questions
are motivated by other works, including those by Curry and his coauthors, that
aim at understanding the dynamics of a rational map $f:
\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}$ restricted to its Julia set.
