District
TheocBase/TheocBase
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District
District
 
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This was previously translated in Dutch as province: part of a country that can have many city and towns. The help-text now suggests that what is meant here may be part of a city.
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I previously translated the same in French... But now with the comment, I will translate it as "Quartier" (Nachbarschaft in Deutch?)
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Yes, it is a part of a city (e.g. Amsterdam-West), my guess for Dutch is "stadsdeel"; in German (Deutsch) it is translated "Stadtteil".
I made some tests in the "Add address" dialog (Territories>Addresses), but in French speaking areas it was rarely filled. Maybe the examples I found in Canada help to decide what fits better : "La Cité-Limoilou" in Ville de Québec and "Le Plateau-Mont-Royal" in Montréal are both "arrondissement", but in Paris for instance the city field contained this information and the "district"-field was empty. 
History
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DistrictDistrict
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DistrictDistrict
District
 
Sublocality, first-order civil entity below a locality