I have a lease for my apartment in NC. I emailed my landlord and ask is I could break my lease. My landlord emailed back with a reply of yes and that I would need to pay a fee equal to two months rent. A day went by and the landlord emailed again stating that he wanted to take back that offer and I could not break my lease. Can the landlord do this, given, that the landlord already made a offer typed to me in email.

Welcome to AMHD. He offered to release you. Did you accept before he withdrew the offer? Is your acceptance of his offer in writing? If not, review the terms of your lease.

The offer was via email and the email also stated for me to take time to review the offer.

The offer seems to have been withdrawn. See: "Offer and acceptance analysis is a traditional approach in contract law used to determine whether an agreement exists between two parties. As a contract is an agreement, an offer is an indication by one person (the "offeror") to another (the "offeree") of the offeror's willingness to enter into a contract on certain terms without further negotiations. A contract is said to come into existence when acceptance of an offer (agreement to the terms in it) has been communicated to the offeror by the offeree."
Offer and acceptance - Wikipedia, the free encyclopedia

Well, this is a lease agreement. It would be under rental- tenant landlord laws and you quoted wikipedia, which is an opinion based website. I need a more provable source

Hello law:

While an offer is on the table, and (1) you haven't accepted it, (2) you haven't made a counteroffer, or (3) you haven't asked them to hold the offer open (which would involve consideration), they can withdraw the offer.

Let's say you were selling a car, and somebody came by and said he'd buy it, but DIDN'T give you a deposit, and went home to get the money.. A minute later, some guy walks up with CASH. Are you obligated to "hold" your deal for the first guy - who may or may not come back??

No, of course not. Now, had he made an offer AND given you some form of consideration (a check), then you would have to wait.

This is the same thing. Their offer is NOT open ended, unless you PAY for it to be so.

excon

Both George and excon are 100% correct. There are no landlord/tenant laws that apply to this situation. This is basic contract law. If someone makes an offer, that offer can be revoked at any time up until the other person accepts the offer and pays the consideration. If you had handed the landlord the two months rent before the landlord withdrew his offer then he would've had to honor it. But he withdrew it before you accepted, which he was legally entitled to do.