Finished near the start of December (6)
I believe the answer is:
closed
'finished' is the definition.
(closing is a kind of finishing)
'near the start of december' is the wordplay.
'near' becomes 'close' (similar in meaning).
'the start of' indicates taking the first letters.
The first letter of 'december' is 'd'.
'close'+'d'='CLOSED'
(Other definitions for closed that I've seen before include "Exclusive" , "restricted" , "Held in private" , "clinched the sale" , "Not open or available to all" .)