At the first mayoral debate earlier this week, Mayor Bloomberg was booed when he said that Rudy Giuliani would be a good governor of New York. That’s because when he swung hard to the right in the presidential election, he alienated the liberal city-dwellers who had generally positive feelings for him as he left office after September 11. But Hizzoner is banking that there are still parts of the city that have good feelings about his predecessor, so he’s asked Giuliani to help his mayoral campaign. It’s a gamble — it may energize some voters, certainly. But as Bill Thompson is running ads that repeat the word “Democrat,” is it really the best idea to keep reminding this blue-dyed city that you’re running as a Republican?