Appearing in downtown Austin on Wednesday with former Hewlett-Packard CEO and prominent Republican Carly Fiorina, GOP gubernatorial candidate Greg Abbott said women in Texas would keep the state red.

Campaigning with Fiorina is Abbott's latest attempt to appeal to women voters in Texas, a demographic with which Abbott held a slight lead over Democrat Wendy Davis in a recent Texas Lyceum Poll. The poll showed Abbott leading Davis 46 percent to 44 percent among female voters, though that difference was within the survey's margin of error. The poll showed Abbott leading 49 percent to 40 percent overall.

"I am winning the women's vote in the state of Texas," Abbott said at a campaign-organized women's lunch at the Four Seasons. "It will be Texas women who keep Texas red."

Davis, a state senator who shot to national fame with a filibuster against a bill restricting abortions, is also fighting for votes from women. She has repeatedly hit Abbott for his stance on abortion and equal pay, pointing to pay disparities in Abbott's own office. And this week, she is launching a radio ad featuring First Lady Michelle Obama, the Houston Chronicle reported.

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“I truly do not understand how women can support his candidacy,” Davis said on MSNBC earlier this week. "His positions on [abortion] are not in keeping with the positions of most people in our state."

Earlier this summer Fiorina launched the Unlocking Potential Project, which aims to recruit conservative women to educate other voters about elections and issues. On Wednesday, Fiorina, who was the Republican nominee for a U.S. Senate seat in 2010 and has declined to say whether she'll run for office again someday, outlined her rise from secretary to chief executive at Hewlett-Packard — a position she was forced to resign from in 2005.

Fiorina spoke out against California's economy Wednesday and said Abbott will keep Texas' economy strong. 

"California is the test case, California is the cautionary tale of what happens when liberal policies are in place for too long," Fiorina said. "The truth is — men and women have done badly under liberal policies."

Abbott and Fiorina touted the number of businesses owned by women in Texas — the second most in the nation. According to the 2013 State of Women-Owned Businesses Report, California has the most businesses owned by women in the country.

"My goal as governor is to make Texas No. 1 in the nation for women-owned businesses — the women in this room can make that happen," Abbott told the audience, which included men and women. 

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In September, Abbott announced he wanted to increase state funds for women's health by $50 million, and expand services to women who are sexually assaulted or have postpartum depression. Abbott also ran ads featuring his mother-in-law endorsing him. 

After the Austin event, Abbott left for another luncheon with Fiorina in Dallas.  

Meanwhile on Wednesday, Davis visited her campaign’s Get Out the Vote training in Fort Worth and addressed her field team, preparing for the last week of campaigning before early voting begins on Oct. 20.