Sorry, You are incorrect. Maybe a little history will clarify. The entire island of Ireland is historically divided into 4 provinces (Ulster, Leinster, Munster and Connaught). Each province contains a number of counties. The counties in Ulster are Donegal, Derry (otherwise Londonderry), Antrim, Armagh, Tyrone, Fermanagh, Down, Cavan and Monaghan. When British rule over all of the island of Ireland ceased circa 1920, with the formation of the Irish Free State, six of the above counties (Derry, Antrim, Armagh, Tyrone, Fermanagh and Down) were retained, and still are, under British rule. The remaining three counties, including Donegal, became part of the Irish Free State, which was subsequently renamed the Republic of Ireland (or Eire in Gaelic). There are twenty-six counties in total in the Republic of Ireland, including the three Ulster counties.
The six counities are generally referred to as Northern Ireland. In pursuing records, those for the six counties are generally held in Belfast, or London, while those for the twenty-six counties of the Republic (including Donegal) are generally held in Dublin.
Hoping this clarifies matters.