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.
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