  The cell complex structure is one of the most fundamental structures in
topology and combinatorics, the Morse decomposition of a dynamical system
analyzes the global gradient behavior, and the Reeb graph of a function is an
elementary tool in Morse theory to represent the global connection and is also
used to analyze continuous and discrete data in topological data analysis. In
this paper, we unify three concepts of cell complexes, Morse decompositions,
and Reeb graphs into a concept. In fact, we introduce topological invariants
for topological spaces and decompositions, which are analogous to abstract
(weak) orbit spaces and Morse graphs for flows. To achieve these, we define
analogous concepts of recurrence and "chain-recurrence" for topological spaces
and decompositions such that the original and new recurrences correspond to
each other for a flow orbits on a locally compact Hausdorff space and the orbit
space. We show that the Morse hyper-graphs exist for any topological spaces and
any invariant decompositions (e.g. compact foliated spaces). Moreover, the
abstract weak element spaces generalize the Reeb graphs of Morse functions and
the Morse (hyper-)graphs and refine abstract cell complexes.
