If I were to sell a piece of property to someone but they didn't know if they wanted it so I agreed in writing to keep the offer open for 1 year could that person sue me if I were to sell the property to someone else who offered me more before the year was up.
It sounds as if you are referring to an option contract. An option contract is a promise made by an offeror that effectively limits the offeror's power to revoke. It specifies a fixed period within which the offeree must exercise the option and communicate acceptance to the offeror to avoid the lapse of the offer.
If one party gives consideration to the other party, the offer cannot be revoked. So, in your example, if you were the offeror and the offeree had paid you a deposit or other ...
Breach of contract is depicted.