yes you should ask for this money because this will be your liability
These customers are entitled for unconsumed part of the year if you close down the gym, however if you plan to continue running the gym (whatever name) then you are liable to provide these customers services till their year completes

Funny.. I made a video about this not long ago:
https://youtu.be/AXVpEh7ZHvc

The answer is.. It depends on what you negotiate.
The prepaid memberships should be on the balance sheet as 'deferred revenue.'
It's a debt to customers for undelivered service but it rarely appears because most gyms just record each transaction as a sale.

Cheers

Dave

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No, unless you keep the same name and entity of the business.

Good day.

I think that this is dependent to the details of your contract agreed upon to both parties.