Rep. Anthony Weiner (D) is keeping hope alive — he is refusing, for now, to back away from the 2009 mayoral election, even though the billionaire incumbent, Michael Bloomberg (I), has changed the city’s term limit law so he can run again.

According to several New York media outlets, Weiner held fundraisers in all five boroughs of the city on Sunday, calling the events “a conversation about how we can restore democracy in New York City and make sure that middle-class families and those aspiring to get there have a real voice in their city’s future.”

Since almost winning the Democratic mayoral primary in 2005, Weiner was seen as certain to try again this year. But his candidacy is less of a sure thing now that the term-limit law has been lifted, and Bloomberg is poised to spend tens of millions of dollars to win a third term.

New York City Comptroller William Thompson has already entered the Democratic primary for mayor, and City Councilman Tony Avella (D) is also planning to run.

Other than his intention to run again, Bloomberg hasn’t said much about his plans. A former Democrat who switched parties to become a Republican for the express purpose of running for mayor in 2001, he was elected twice on the Republican line but became an Independent in 2007.

This year, he will certainly run as an Independent, but he could try to win the Democratic and Republican lines — and the nominations of minor parties as well — with the permission of local party leaders.

On the Republican side, supermarket magnate John Catsimatidis has created an exploratory committee for a possible City Hall bid but has yet to formally enter the race.