  In this paper we define unstable topological entropy for any subsets (not
necessarily compact or invariant) in partially hyperbolic systems as a
Carath\'{e}odory dimension characteristic, motivated by the work of Bowen and
Pesin etc. We then establish some basic results in dimension theory for Bowen
unstable topological entropy, including an entropy distribution principle and a
variational principle in general setting. As applications of this new concept,
we study unstable topological entropy of saturated sets and extend some results
in \cite{Bo, PS2007}. Our results give new insights to the multifractal
analysis for partially hyperbolic systems.
