Council Tax

Other

Dear sir
tenancy was periodic with a notice period of one month required. Tenant gave notice on 12 December 2020 via email stating that he would vacate 14th January 2021. He duly paid the final months' rent. The keys were returned to me on 20th December 2020 but the local authority have accepted his assertion that he left the property on 12 December 2020. A new tenant moved in on 14th January 2021 but the authority are seeking one day of council tax from me. Are they within their rights on the basis that the tenancy formally ended on the 14th January, rent was paid to this date and they have had sight of the email from the tenant proving the case?
I know this is a very small amount of money but would not want to accept a situation where a tenant was empowered to entrap a landlord into liability for council tax.
Thank-you