Unaware of endless falsehood in open (9)
I believe the answer is:
oblivious
'unaware of' is the definition.
(I've seen this before)
'endless falsehood in open' is the wordplay.
'endless' means to remove the last letter.
'falsehood' becomes 'lie' (lie is a kind of falsehood).
'in' indicates putting letters inside.
'open' becomes 'obvious' (similar in meaning).
'lie' with its last letter taken away is 'li'.
'li' placed into 'obvious' is 'OBLIVIOUS'.
(Other definitions for oblivious that I've seen before include "Heedless" , "Blind (to)" , "Apt to forget" , "Not noticing" , "like Lethe?" .)