Legal Question in Landlord & Tenant Law in New York

If you accepted rent and there is no written agreement, then you probably have a month-to-month tenant, who has to have full month's notice of termination. You serve a notice sometime this month. Sooner is better than later even though it makes no legal difference. The notice says that the notice is given as of October 1, 2013 and the tenant's lease is terminated as of October 31, 2013. If the tenant does not remove then you commence a proceeding for such removal. The notice has to say all of that, be signed by you, and served in the same manner as a summons. On November 1, you go to Town Court in Hamburg and file a summary holdover proceeding, serve the petition & notice of petition, appear in court on the scheduled date and try the case.

If you have not gotten rent, how is this a tenant as opposed to a mooch or trespasser?