The class of separable Banach spaces with the π-property along an analytic filter is claimed to be Σ^1_3, and Σ^1_2 for countably generated filters, but the proof as written has a critical error.
Ghawadrah, The descriptive complexity of approximation properties in an admissible topology
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The $\pi$-property of a Banach space along a filter
The class of separable Banach spaces with the π-property along an analytic filter is claimed to be Σ^1_3, and Σ^1_2 for countably generated filters, but the proof as written has a critical error.