  The unrivaled robustness of topologically ordered states of matter against
perturbations has immediate applications in quantum computing and quantum
metrology, yet their very existence poses a challenge to our understanding of
phase transitions. However, a comprehensive understanding of what actually
constitutes topological order is still lacking. Here we show that one can
interpret topological order as the ability of a system to perform topological
error correction. We find that this operational approach corresponding to a
measurable both lays the conceptual foundations for previous classifications of
topological order and also leads to a successful classification in the hitherto
inaccessible case of topological order in open quantum systems. We demonstrate
the existence of topological order in open systems and their phase transitions
to topologically trivial states. Our results demonstrate the viability of
topological order in nonequilibrium quantum systems and thus substantially
broaden the scope of possible technological applications.
