Democrats still have the edge. "The economy is the more compelling issue in this election, but the difference is that Romney only has the economy. Obama has the economic argument, but he has the other issues on top of that," said Susan Carroll, a senior scholar at the Center for American Women and Politics, a nonpartisan research organization in New Jersey.

To win the White House, Democrats need to maintain their recent edge with women, particularly in an election that promises to be close. In every White House race since 1980, the Democratic candidate has done significantly better with women than men.

Romney got some fresh hope from an Oct. 4-7 Pew Research Center poll showing him tied with Obama among women. In September, Romney was down 18 percentage points.

Michael Dimock, the center’s associate director, said the change could have been fueled by Romney not coming off in the first debate as the monster Obama had portrayed.

"The expectation was that people would see this aristocratic, out-of-touch ogre on the stage," Dimock said.

Evidence remains strong, though, that Romney still has an enormous task. "Do I think there was some movement after the first debate? Absolutely,” Carroll said. “But I don’t believe it completely reversed it (the gap)."

The Marist College Institute for Public Opinion has polled voters this month in five swing states, and director Lee Miringoff saw the gender gap persisting.

Nor have the debates appeared to change many minds, he said. "Voters aren’t shopping around right now," he said.

The Obama-Romney choice for women depends on which issues they view as more crucial. A Politico-Battleground poll Oct. 7-11 asked which issues were most important. The economy and unemployment were overwhelmingly the top choices.

Gallup, though, reported this month that 39 percent of women considered the economy the most important issue for women in this election, while 15 percent cited equal rights and pay. Those issues, said Gallup’s Andrew Dugan in a poll analysis, "play to Obama’s strengths."

The size of the gender gap, he said, "may rest in part on the Obama campaign’s ability to emphasize these issues as especially critical to the female electorate."

Romney, on the other hand, "must continue to focus on his strengths on more general national issues," such as the federal debt and the economy, "and continue to highlight their importance to women."