President Obama campaigned in this essential swing state Wednesday with a special appeal to female voters, warning a raucous audience in a basketball gymnasium here that Republican challenger Mitt Romney wanted to take women’s health rights “back to the 1950s.”

To shouts of support, Obama spoke proudly of Obamacare, the name Republicans once derisively labeled his health-care reform legislation. On Wednesday he acknowledged to many of its beneficiaries here that “I kind of like the name.”

He listed a number of the law’s benefits — calling it “Obamacare” each time — and showed the kind of unabashed liberal pride in achieving universal health coverage that his base has longed for.

Obama again framed the election, which his aides acknowledged again only hours earlier will be very close, as a choice between progress he said that he has helped oversee and retrenchment in areas such as education and women’s reproductive rights.

“There’s nothing conservative about a government that allows a woman to make her own health-care decisions . . . he [Romney] says he’s the candidate of freedom. But freedom is about making decisions about your own health care and when you need it.”

“We are not going backwards, Denver, “ he continued, “we are moving forward.”

The event here began a two-day swing through Colorado, including to several regions that voted against Obama in 2008.

But his political weakness among working-class whites, here and in the handful of other swing states that will decide the election, underscores the importance of widening his lead over Romney among such key constituencies as women and Hispanics, whose growth has been rapid here.

To huge applause, Obama listed the impending anniversaries of Sonia Sotomayor and Elena Kagan joining the Supreme Court, his two nominees so far.

Obama spoke in personal terms about his wife, Michelle, saying that it was important to him that her rights to equal pay and health care are protected. And he said that he wants his daughters, Malia and Sasha, to have the same opportunities as anyone’s sons.

“Thanks to you, we’ve made a difference in people’s lives,” Obama said, referring to people he has met in the past two years who have benefited from his health-care act.

But Obama faces political challenges in Colorado, despite the fact he has a lot going for him here politically.

The state is young (median age 35.5), nearly 20 percent Hispanic and is packed with Democratic newcomers who have transformed a historically Republican stronghold into a bona fide battleground. Those were among the factors that helped him win by a nine-point margin, 54 to 45 percent, in 2008.

But the Colorado where the president will campaign over the next two days poses new challenges. A Quinnipiac University/CBS/New York Times poll of three swing states released Wednesday shows Obama trailing Romney, 50 to 45 percent, among likely voters.

His advantage over the former Massachusetts governor with women (51 to 43 percent) is significantly smaller than in Virginia (54 to 40 percent) or Wisconsin (59 to 36 percent), the other two states covered in the poll. His deficit in Colorado among men (Romney has a 56 to 39 percent edge) is larger.

There also appears to be a softness in Obama’s support in Colorado: 11 percent said they might change their minds before Election Day, compared to 6 percent for Romney.

On the economy, the news for Obama is just as troubling. Romney enjoys a 10-point edge (51 to 41 percent) on the question of who would do a better job with the economy. In Virginia and Wisconsin, the two candidates are running essentially even on the issue, the poll found.

Colorado is perhaps more difficult this year because of an electorate rocked by back-to-back traumas, neither of which was within Obama’s control: the Aurora mass shootingand a series of ruinous wildfires. Obama told the audience Wednesday that the shooting victims and their families are still in the nation’s thoughts.

Obama will spend the next two days on the ground in Denver, Grand Junction and Pueblo, talking about economic and women’s issues.

The Obama campaign said the president has visited Colorado nine times since taking office.

Obama is also expected to continue hammering Romney’s plan to cut taxes on the wealthy. (Sen. Rob Portman (R-Ohio), widely viewed as a possible vice presidential pick, was also in the state Wednesday, campaigning for Romney.)

Seth Masket, a political science professor at the University of Denver, said Obama appears to be taking a page from the 2010 Senate campaign of Michael Bennet, who narrowly defeated tea party-backed Republican Ken Buck. In his primary campaign, Buck alienated women with a series of statements, including urging voters to choose him over opponent Jane Norton because he “doesn’t wear high heels.”

“He’s somewhat following the Bennet playbook,” said Masket. “Obama is trying to use that strategy and is hoping for a similar result.”

Masket and other analysts say Obama’s success will depend on the ability of his ground-level organization in the state to drive turnout, as it did successfully in 2008.

He estimates that the Obama operation has about 30 field offices spread across the state, including some in conservative areas the campaign is not likely to carry — Colorado Springs and Grand Junction — but where there are still sizable numbers of Democrats to be targeted.

But Romney is also ramping up, hoping to duplicate Obama’s 2008 success.

“Colorado is a key state for us, we have a great ground game here, and we know we can’t leave one stone unturned in trying to win as many votes as possible,” Jennifer Psaki, the Obama campaign’s press secretary, told reporters Wednesday aboard Air Force One.

The ground game will be ever more important this year. Coloradans, who have been marinating in television ads from both sides, say they have tuned the spots out. According to the Quinnipiac poll, 67 percent of likely voters say the ads are “not important” to their decision making.

To chants of “four more years, four more years” in the venue here, Obama responded with a smile, “I’ll tell you what — if we win Colorado, we’ll get four more years.”

At the event here focusing on women’s health issues, Obama was introduced by Sandra Fluke, a Georgetown University Law Center student who earlier this year was called a “slut” and a “prostitute” by the conservative radio talk show host Rush Limbaugh for her advocacy in favor of insurance mandates covering contraception. She was also blocked by House Republicans from testifying in favor of coverage for contraception

“When I was attacked earlier this year verbally, the difference between Mr. Romney and President Obama was very clear,” Fluke said, describing Obama’s public defense of her statements.

“Mr. Romney could only say those weren’t the words he would have chosen,” Fluke said to cheers. “Well, Mr. Romney, you are not going to be the candidate we choose.”