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We’re fact-checking the debate in real time.

— Mr. Trump

While it is true that Mr. Trump had vacancies to fill when he assumed the White House, the reason is not simply that former President Barack Obama “left” the positions vacant. The Republican-led Senate refused to confirm many of Mr. Obama’s judicial nominees, including Judge Merrick Garland, whom Mr. Obama named to fill the vacancy left by the death in February 2016 of Justice Antonin Scalia of the Supreme Court.

— Mr. Biden

Mr. Trump’s former counselor, Kellyanne Conway, told Fox News in August that Mr. Trump would benefit politically from unrest in American cities.

“The more chaos and anarchy and vandalism and violence reigns, the better it is for the very clear choice on who’s best on public safety and law and order,” she said.

— Mr. Trump

Mr. Trump is taking undue credit for the relative calm that has settled in Minneapolis, a city roiled by protests in May. The governor, not the president, sent the National Guard there. Gov. Tim Walz of Minnesota activated the state’s National Guard on May 28, three days after George Floyd’s death. The guard tweeted at about 4 p.m. local time that it was ready to respond to the governor’s request.

Mr. Trump tweeted around midnight telling Mayor Jacob Frey of Minneapolis to “get his act together” or “I will send in the National Guard & get the job done right” — an hour after the state National Guard said it had deployed 500 members to the city.

Seattle Mayor Jenny Durkan told the Washington Post, that a conversation that Mr. Trump described “just never happened.”

— Mr. Biden

The trade deficit with China — the gap between what America exports to China and what it imports — fell sharply between 2018 and 2019 as Mr. Trump’s trade war suppressed commerce between the world’s largest economies.

So far this year, the trade deficit in goods with China is running below last year’s levels, as the United States imports fewer products. But while Mr. Trump has lowered the trade deficit with China, deficits with other countries have grown and the overall trade deficit is once again trending up.

American consumers have shifted to buying more goods from countries like Vietnam and Mexico, in part because Mr. Trump’s tariffs on $360 billion worth of Chinese goods have raised prices on those imports.

The overall United States trade deficit shrank last year for the first time in six years. So far this year, it is once again climbing sharply, as both imports and exports remain below pre-pandemic levels. The monthly trade deficit in July hit the highest level in 12 years, and the trade deficit in goods with Mexico hit a new record.

— Mr. Trump

This claim is based on an investigative report released last week by Senate Republicans that accused members of Mr. Biden’s family of cashing in on his vice presidency. The report claims that Hunter Biden “had a financial relationship” with Elena Baturina, a wealthy Russian businesswoman and the widow of a former mayor of Moscow. The report based this claim on an unidentified “confidential document” showing that Ms. Baturina transferred $3.5 million in 2014 for “a Consultancy Agreement” to a bank account associated with a company called Rosemont Seneca Thornton, which was associated with Hunter Biden’s business partners.

Hunter Biden’s lawyer has said that he was not a co-founder of Rosemont Seneca Thornton, had no interest in it and did not have a financial relationship with Ms. Baturina. He did not respond to a question about whether Mr. Biden was paid by Rosemont Seneca Thornton or did consulting work for Ms. Baturina.

— Mr. Trump

Airport hangars cannot accommodate crowds of that size. While Mr. Trump’s rallies in the past have attracted tens of thousands of attendees, in recent weeks the rallies that he has been holding at airport hangers have been far smaller. According to local news reports this month, a rally at an airport in Virginia drew an estimated 3,000 people, an airport rally in Michigan drew an estimated 10,000 people and a rally in Pennsylvania drew an estimated 7,000 people. Mr. Trump has a tendency to exaggerate his crowd sizes, starting with his inauguration in 2017.

— Mr. Trump

Mr. Trump is right that the growth rate of economic output as measured by gross domestic product was slower after the recession that spanned 2007 to 2009 than it had been following other contractions.

But that fact is misleading in isolation. Growth had been slowing for decades as the population aged and other long-run trends caused the economy’s potential run rate to decline. Nor did growth pick up dramatically once Mr. Trump took office, aside from a short-lived jump on the back of his tax cuts.

It is worth noting that the 2007 to 2009 recession was the worst since the Great Depression, and its depth and length led to labor market scarring, which trapped many would-be workers on the sidelines of the job market.

— Mr. Biden

As of mid-September, 1 in 1,020 Black Americans has died of Covid-19 — the highest rate of death when broken down by race and ethnicity. Since the early days of the pandemic, the coronavirus has disproportionately affected Black, Latino, Native, and Indigenous people, who are contracting the virus at higher rates and are more likely to be hospitalized for severe Covid-19.

1 in 1,220 Indigenous Americans, 1 in 1,400 Pacific Islander Americans, and 1 in 1,540 Latino Americans have died from the virus, compared to 1 in 2,150 white Americans and 1 in 2,470 Asian-Americans. Such statistics are supported by data from the Centers for Disease Control and Prevention, which note that people who identify as Black or African-American are more than twice as likely to die from the coronavirus, compared to their white neighbors.

— Mr. Trump

Mr. Biden’s son, Hunter Biden, is involved in a Chinese government-linked private equity fund, BHR Equity Investment Fund Management Co., that won a business license from the Chinese government. Hunter Biden was on the board of the fund when it was formed in late 2013, and he later invested roughly $420,000, giving him a 10 percent stake, after his father had left the vice presidency.

But Hunter Biden’s lawyer has said that he has never been paid for his role on the board, and has not profited financially since he began as a part owner. Hunter Biden left the board in April, according to a letter produced by his lawyer. But as of June, he still owned his stake in the fund, which he was trying to sell. His lawyer did not respond to a request for comment about the status of that effort.

— Mr. Trump

President Trump publicly pressed the Big Ten to reverse its decision not to play football this autumn, and he even spoke to the league’s commissioner, Kevin Warren. But Big Ten officials, who voted this month to try to play beginning in October, insisted that they accepted no federal aid and that Mr. Trump was not a pivotal figure in the league’s deliberations.

— Mr. Biden

Mr. Biden may be relying on the Bureau of Labor Statistic’s nonfarm payroll survey, which stretches back to the late 1930s, to arrive at this conclusion. But Herbert Hoover, who was president during the Great Depression, left office in 1933 at a time when the economy had fewer jobs than when he was elected in 1929, based on subsequent estimates. Mr. Biden’s statement also requires the unproven assumptions that Mr. Trump will lose the election, and that jobs will not bounce back to pre-crisis levels before November.

— Mr. Trump

Mr. Trump claimed his rallies have had “no negative effect” because of the coronavirus and that as many as 35,000 or 40,000 people have attended the events. Both are untrue, as is a separate claim that his rallies have all been held outdoors.

At least eight campaign staff members who helped plan President Trump’s indoor rally in June in Tulsa, Okla., including members of the Secret Service, tested positive for the coronavirus, either before the rally or after attending.

Mr. Trump’s rallies have generally attracted just several thousand people, not the tens of thousands he claimed. While the president’s campaign had claimed that more than 1 million people had sought tickets for the Oklahoma rally, the 19,000-seat arena was at least one-third empty during the rally. A second, outdoor venue for an overflow crowd at the same event was so sparsely attended that he and Vice President Mike Pence both canceled appearances there.

— Mr. Trump

Mr. Trump did not “bring back” 700,000 manufacturing jobs, even before the coronavirus recession. In his first three years as president, manufacturing employment rose by just under 500,000 jobs. Through August, because of jobs lost to the pandemic recession, the sector it is down by more than 200,000 jobs from when Mr. Trump took office.

— Mr. Biden

Mr. Biden missed the point. Mr. Trump’s taxes reveal he does take advantage of deductions and tax credits available to him. But the main reason he does not pay income tax is because his businesses lose far money than they make.

Because of the way the tax code works, businesses can use losses in one year to avoid paying income tax in future years. Mr. Trump has no shortage of losses. Take Trump National Doral, his golf course near Miami. Mr. Trump bought the resort for $150 million in 2012. Through 2018, his losses have totaled $162.3 million.

Overall, since 2000, Mr. Trump has reported losses of $315.6 million at his golf courses. And his namesake hotel in Washington, D.C., showed losses of $55.5 million through 2018.

— Mr. Trump

The president was referring to the relative risks to young people from the coronavirus. The vast majority of children do not become visibly ill when infected with the coronavirus. But while a strong immune system may protect them from becoming sick, they are far from immune. Several studies have shown that children can get infected and harbor high levels of the coronavirus. And a small proportion of children seem to develop a condition called multisystem inflammatory syndrome, a severe and sometimes deadly overreaction of the immune system.

The debate on schools has mostly centered on whether children who are infected can transmit to others. The bulk of the evidence here suggests that children under 10 are about half as likely to spread the virus to others, but older children, particularly 15 and above, may transmit the coronavirus as efficiently as adults do. Teenagers are also about twice as likely as younger children to be infected with the coronavirus, according to a new report from the Centers for Disease Control and Prevention, suggesting that high schools and colleges may be important contributors to community spread.

— Mr. Biden

Scientists projecting the death toll of the virus in the United States have noted that, should the country maintain its current levels of physical distancing mandates and masking, more than 370,000 Americans could be dead by January 1, 2021 — about 165,000 more than the current death toll.

Should masking and distancing become very widespread, as Mr. Biden references, the total death count would be around 275,000, potentially saving nearly 100,000 lives. Models have also projected potential deaths if mandates were to ease, allowing further mingling and exposure. Eased mandates could catapult the country onto a path toward reaching 425,000 deaths by January of next year.

— Mr. Trump

While Mr. Trump appears to have paid a variety of taxes in recent years, including payroll taxes for his employees, he has paid very little in federal income taxes, according to tax documents obtained by The New York Times.

They show that in 2017, for example, Mr. Trump chose to pay $750 in federal income taxes. That was the case even though he reported earning some $15 million for the year, through a variety of sources. But on his federal tax return, Mr. Trump offset those earnings by reporting losses from his businesses and claiming a range of tax credits, including one that allowed him to reduce his liability under the alternative minimum tax from $7.4 million to $750. It is unclear how his accountants chose that number: Mr. Trump appeared to have sufficient credits to reduce his liability to zero. That same year, Mr. Biden paid about $3.7 million in federal income tax, his returns show.

— Mr. Trump

Mr. Biden, who has stressed the importance of following scientific expertise in responding to the pandemic, is not promising to shut down the whole economy.

In an interview with ABC News in August, Mr. Biden was pressed on what he would do “if the scientists say shut it down” and did respond, “I would shut it down. I would listen to the scientists.”

But this month, Mr. Biden said, “There is going to be no need, in my view, to be able to shut down the whole economy.”

— Mr. Trump

Mr. Trump often claims that his administration had fostered the best economy in history before the onset of the pandemic. But data show that the expansion that he presided over — which he inherited — failed to measure up to prior economic eras across several dimensions.

The expansion from 2009 through early 2020 was the longest on record. It saw years of strong labor market gains that pushed the unemployment rate steadily lower, until it hit 3.5 percent and held around that half-century low for much of 2019 and early 2020. The robust labor market led to stronger wage gains for low earners and helped to fuel consumer spending.

But many people remained on the job market’s sidelines: the employment rate for men in their prime, for instance, never rebounded to pre-crisis levels.

Output growth, which did receive a temporary boost from Mr. Trump’s tax cuts, has otherwise generally oscillated around 2 percent. That is roughly the level economists see as sustainable given modern productivity and demographic trends, and lower than the run rate that prevailed in prior decades.

And inequality remained very high. The top 1 percent hold almost 40 percent of the nation’s wealth, based on a Federal Reserve survey, while the bottom 50 percent of wealth-holders had only about 1 percent of the overall pie. Those 2019 figures are little changed from 2016, Fed economists said.

— Mr. Trump

Top health officials have said that a vaccine may not be widely available until next summer. Dr. Moncef Slaoui, the top scientist on the administration’s vaccine development program, recently said that Americans would most likely not be widely vaccinated until the middle of 2021, a timeline echoed by Dr. Robert R. Redfield, the director of the Centers for Disease Control and Prevention. Dr. Slaoui also said that the chance of having a vaccine by October or November was “extremely unlikely.”

Of the companies with vaccines in late-stage clinical trials in the United States, just one — Pfizer — has said that it could have initial results by the end of October, a time frame the company has clarified is a best-case scenario.

At the same time, Dr. Anthony Fauci and other top health officials in the administration have said that there could be evidence of a vaccine’s effectiveness by November or December. If every aspect of the vaccines’ development and distribution goes exactly as planned, certain people in high-risk groups, including frontline health workers, could get vaccinated this year.

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