The Southern District Court covered Manhattan, Bronx and Westchester counties.  However, people were free to file in whichever court was most convenient for them, so they might have done so if they worked nearby, or if the court calendar fit their schedule better.

The naturalization records from this court for the period 1824-1945 are available online for free (registration required) from the comfort of your home:  https://www.familysearch.org/search/catalog/51903 .  Using the petition number that you obtained from the index, scroll down the catalog page until you locate the corresponding "roll", then click on the camera icon.

Regards,
David Oseas

Researching:
HYMAN/HEYMAN/HEIMOWITS/CHAJMOVITS: Zemplen-Dobra, Hungary > New York
KLEIN: Satoraljaujhely (Ujhely), Hungary > New York > Los Angeles
KRONOWITH: Hungary > New York
OSEAS/OSIAS/OSIASI/OZIAS: Iasi, Romania > Chicago > Milwaukee > Los Angeles
SCHECHTER/SHEKTER: Kishinev, Bessarabia > New York  
SHERMAN: Iasi, Romania > New York > Los Angeles
STRUL:  Iasi, Romania > Haifa, Israel
WICHMAN: Syczkowo (Bobruisk), Belarus > Milwaukee > Los Angeles