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But Weiner said the campaign was "too important to give up because I've had embarrassing personal things become public." And he said he wasn't surprised his opponents wanted him out.
Democratic strategists in New York and Washington, where Weiner served seven terms before resigning in 2011, said there are few external means of pressuring Weiner to drop out.
Weiner has nearly $5 million to spend on the campaign, allowing him to mount a vigorous defense on television. Also, he was not particularly close to his colleagues in the congressional delegation, the strategists said, so he might be unmoved if they urge him to exit the race.
As for the voters, some want Weiner to go.
"He's disgusting," said magazine editorial assistant Katelin Marinari, 24.
But others said they would still vote for him.
"Do I think morally he's wrong? Of course. But I'm not voting for a minister; I'm voting for a mayor of New York," said public relations worker Raven Robinson, 22.
A Quinnipiac University poll released Wednesday continued to place Weiner in the top rung of Democratic contenders with Quinn and Thompson, though it was taken largely before the scandal broke.
"I have posited this whole campaign on a bet, and that is that, at the end of the day, citizens are more interested in the challenge they face in their lives than in anything that I have done, embarrassing, in my past," he told reporters outside his Manhattan home Wednesday morning.
The unidentified woman involved in the newly disclosed messages told The Dirty that she was 22 when she began chatting with Weiner on a social networking site in July 2012, and that their exchanges lasted six months.
The Dirty posted explicit conversations of two people fantasizing about various sex acts, and ran a pixelated photo of what it said were Weiner's genitals.