Can a live in Nanny be sued for not giving 2 weeks notice quitting?

The starting point of all employment relationship law in Texas is what is known as at-will employment.  Under this doctrine both the employee and the employer can terminate the employment relationship with or without notice for any reason or no reason at all. There is nothing in the law about either party having to give the other two weeks or any other notice of the end of the relationship. That said, the parties can agree to a required notice period.  Also note the law may create a promise where none is expressly given if one party relies to its detriment on the representation.  Example:  Employer says to a hypothetical Nanny "We are planning to go to Europe next weekend, will you be available to take care of the kids next week?" and the Nanny says "sure" upon which the Employer buys non-refundable plane tickets.  Nanny quits the next day without notice.