Penalty points on british license

[Briot1]

Driving from Dublin to Waterford I was stopped just outside of Carlow (which I now hear is notorious) by an unmarked police car. They said I was doing 140 kph. which I admit is very dangerous.. However, they advised me to slow down, and let us leave. .within about half an hour, we noticed we were being followed for about 5 miles by a Garda car , I was not speeding, as there was another car in front of me and I was being extremely cautious, however once that car moved, the garda pulled in front of us and directed me off the main road, I had my little boy in the car and I was worried, I know I wasn't doing anything wrong The Garda was very belittling, constantly going on about 'how Irish roads aren't like English roads' and disbelieving that I lived in the UK, he was nasty he kept us for about half an hour going back and forth to his car, my friend was so upset its put her off ever going to Ireland again. anyway, moving on, I received a fine which I paid, now I've had a letter from the Irish RSA saying they will apply 3 points to my license, how can they do this? and can I appeal?