*In other words, what restrictions on a family F of {0,1} valued borel functions will tell us that the pointwise limit of any net in F is borel. I feel like there must be lots known about this but I cannot seem to find anything.
When is a subset of {0,1} valued borel functions on a standard borel space (polish space) complete (see *) under the pointwise convergence topology?
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analysis
topological-vector-spaces
descriptive-set-theory