Bravery in taking Australian top dollars to any open cosmopolis (8)
I believe the answer is:
audacity
'bravery' is the definition.
(audacity is a kind of bravery)
'australian top dollars to any open cosmopolis' is the wordplay.
'australian' becomes 'au'.
'top' suggests taking the first letters (in a down clue, the first letter is top).
'open cosmopolis' becomes 'city' (I am not sure about this - if you are sure you should believe this answer much more).
The first letters of 'dollars any' is 'da'.
'au'+'da'+'city'='AUDACITY'
'in taking' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for audacity that I've seen before include "Effrontery" , "Boldness, may be brazen" , "Courage" , "Brass neck, boldness" , "Temerity" .)