Break up when presented with hollow assertion - they have such outdated beliefs (5)
I believe the answer is:
pagan
'have such outdated beliefs' is the definition.
The answer and definition can be both people as well as being singular nouns.
Maybe there's an association between them I don't understand?
'break up when presented with hollow assertion they' is the wordplay.
'break' becomes 'gap' (a gap is a break in something).
'up' is a reversal indicator (in down clue: letters go upwards).
'when presented with hollow assertion they' becomes 'an' (I can't explain this - if you can you should give a lot more credence to this answer).
'gap' back-to-front is 'pag'.
'pag'+'an'='PAGAN'
Can you help me to learn more?
(Other definitions for pagan that I've seen before include "One who follows a polytheistic or pre-Christian religion, but not a Jew" , "Person without any conventional religion" , "Unbeliever" , "One who does not acknowledge your god(s)" , "Non-mainstream religious believer" .)