Letter of public house, opening for inspection (3)
I believe the answer is:
phi
'letter' is the definition.
(phi is a kind of letter of the alphabet)
'public house opening for inspection' is the wordplay.
'public house' becomes 'ph'.
'opening for' says to take the initial letters.
The initial letter of 'inspection' is 'i'.
'ph'+'i'='PHI'
'of' acts as a link.
(Other definitions for phi that I've seen before include "Greek letter in previous answer" , "Letter from the Balkans" , "Greek letter; sounds like expression of outrage" , "exotic character" , "21st Greek letter" .)