In California, if the tenant does not cure the violation alleged in the three day notice, after three full days the landlord can file an unlawful detainer. If your mother has been a tenant there for at least one year, the landlord can also give her a 60 day [as opposed to 30 days] notice that she has to vacate whether or not the unlawful detainer is successful [assume that she is not under a lease that expires after the 60 days]. Based upon the very limited facts you present, yes the landlord has acted legally. Even if your mother wins on the three day notice, she has to be out after 60 days.
Answered on Jan 04th, 2016 at 2:42 AM