BE02 CREW WAS CLRED FOR TKOF INTO TEMPORARY FLT RESTR AIRSPACE.

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A TEMPORARY FLT RESTR WAS SET UP AT MVY FROM THE 265 DEG RADIAL (MVY) AT 17 DME TO 357 DEG RADIAL (MVY) AT 7.8 DME FOR 10 NM RADIUS AROUND A LINE CONNECTING THOSE 2 POINTS. I WAS CLRED VIA THE 'BRAVO' DEP WHICH IS RWY HDG; RADAR VECTORS TO INTERCEPT THE 017 DEG RADIAL (MVY) FREDO DIRECT BOSTON; CLB 2000 FT.... WE WERE CLRED FOR TKOF. UPON CLBING THROUGH 400 FT; I ASKED MY FO IF WE WERE STILL TO MAINTAIN RWY HDG AND HE SAID; 'NO; R TURN ON COURSE.' I MADE THE TURN AND LEVELED OFF AT 2000 FT AND MVY TWR HADN'T HANDED US OFF TO CAPE APCH YET. WE ASKED TWR IF THEY WANTED US OVER TO DEP AND THEY SAID YES. AFTER CONTACTING CAPE APCH THEY GAVE US A NUMBER TO CALL REGARDING VIOLATING THE TEMPORARY FLT RESTR. MVY ARPT WAS INSIDE THE TEMPORARY FLT RESTR SO IF I WAS IN VIOLATION; THEN EVERY PLT THAT LANDED AND/OR DEPARTED MVY SHOULD BE VIOLATED AS WELL. BOS; LGA; JFK; PHL; AND IAD ARPTS ALL HAD INDEFINITE GND STOPS BECAUSE TFC WASN'T ALLOWED OVER THE AREA; SO I DON'T KNOW WHY THEY DIDN'T CLOSE MVY ARPT. THE INFO SHOULD HAVE BEEN DISSEMINATED BETTER. A LOT OF PLTS ACCIDENTALLY FLEW THROUGH THE AIRSPACE. MVY ARPT SHOULD HAVE BEEN CLOSED! WHEN MVY TWR SAW US TURN; THEY SHOULD HAVE SAID SOMETHING. THEY DIDN'T SAY A WORD AFTER 'CLRED FOR TKOF.'