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The cap-and-trade proposal requires polluters to either reduce their emissions or offset their pollutants by paying others to reduce their emissions.

As they have in previous debates, the two disagreed on closing the state’s so-called “gun-show loophole” that allows people to purchase weapons in such arenas without a background check.

Mr. Deeds said he has supported closing the loophole at the behest of the families of the 2007 Virginia Tech attacks.

Mr. McDonnell said that while he didn’t support the bill, he worked with Gov. Tim Kaine to tighten mental health restrictions for gun sales.

Mr. Deeds said he would renew an executive order ending discrimination based on sexual orientation in the government work force. Mr. McDonnell, as attorney general, issued an opinion declaring a similar executive order issued by Mr. Kaine unconstitutional.

Mr. McDonnell said the law was unconstitutional not because of its social implications, but because it violated the separation of powers.

The debate was widely seen as crucial to Mr. Deeds’ chances of closing a persistent gap in the polls that narrowed last month after news reports about Mr. McDonnell’s thesis but has again showed signs of growing.

A poll conducted last week by Mason-Dixon Polling & Research Inc. for the Richmond Times-Dispatch had Mr. McDonnell ahead 48 percent to 40 percent in the race. The poll of 625 likely voters had a margin of error of four percentage points.

The debate, during which audience members had their questions posed to the candidates, was dubbed the “People’s Debate.” It was sponsored by AARP Virginia and the League of Women Voters of Virginia and held at the studios at WCVE-TV, Richmond’s PBS affiliate.

Judy Woodruff, senior correspondent for “The NewsHour with Jim Lehrer,” moderated the debate, which was nationally televised on CSPAN 3. Attendance at the event was by invitation only.

The fourth and final debate is scheduled for Oct. 20 in Roanoke.