Can a landlord who sent you a termination notice in May make you pay lease payments until the end of November if you vacate the apartment in September?

Answer

If the lease/rental agreement you signed continued into the month of November, then yes you have to pay the charges.

Answer

TIMING and rental agreement is the key here.

Let's suppose that my landlord sends me a termination letter in May to take effect 60 days from the receipt of the letter, then that means that I have up until June 30th to vacate the premises. You will not have to pay the rent for the remainder of the lease because by sending you a termination letter, he is terminating the rental agreement.

Keep in mind that laws vary from state to state, so I would check with the local authorities. I am pretty sure that you can log onto your county's website and search under the tenant rights section.

Make sure you read through your original rental agreement, there you might find a clause that specifies the circumstances under which the tenent or landlord may terminate such agreement.

good luck,

Landlords can be very greedy and one-sided so you must protect yourself. you did not specify how much notice he gave you, but usually tenants get 30 days notice to vacate. Once you get this you are free to leave and not pay any rent beyond the 30 days.
(assuming he had the right to terminate)
BUT, if you remained in the premises you must pay.
So, you did not really give enough information to get a good answer.