When the new school year opens Wednesday in New York City, 65,000 4-year-olds will attend free pre-kindergarten — the largest expansion of public school of its kind in the country.

The pre-kindergarten rolls, larger than the entire K-12 public school enrollment of Boston or Seattle, makes good on a campaign promise by Mayor Bill de Blasio (D) to offer full-day preschool to every 4-year-old in the city.

It also serves as a model for cities and states across the country that are looking to expand preschool in an effort to narrow the achievement gap between poor children and their more affluent peers, he said.

“People are demanding this,” de Blasio said in an interview. “They believe it’s something government can and should do. Bringing all kinds of kids together in a classroom lifts all boats. This is where we have to go. This is where the modern world is taking us.”

“This should be the model going forward all over this country,” said de Blasio, who spent part of the Labor Day weekend visiting barbershops and beauty salons in Brooklyn, talking to parents about enrolling their children in preschool. “If we can do this for 65,000 kids, with all the challenges that New York City has with our rich demographic mix, English-language learners and kids with special needs, I believe you can do it anywhere.”

New York City Mayor Bill de Blasio (Mary Altaffer/AP)

Educators see high-quality ­early-childhood education as especially important to help close the achievement gap, which has been demonstrated to exist among children as young as 3.

Several studies have suggested that low-income children enrolled in high-quality preschool are more likely to graduate from high school and gain lifelong benefits — such as better jobs and higher wages — and are less likely to be in the criminal-justice system and to get social services.

Since 2013, President Obama has been urging a federal-state partnership to fund preschool for 4-year-olds in low-income homes, a proposal that has failed to gain traction among Republicans on Capitol Hill. But an increasing number of Republican governors have moved to expand publicly funded preschool in states including Texas, Georgia, Alabama, Mississippi and South Carolina.

The emphasis on preschool also comes as new brain research is showing that people learn more in their first five years than in any other five-year period.

“Today, pre-K is the equivalent of what kindergarten or first grade was in the past,” said de Blasio, adding that he now considers the country’s largest public school system to be pre-K through 12.

“We believe that with universal pre-K, the entire school system is going to be stronger,” he said.

This is the second year of the preschool program in the city. Last year, about 53,000 children were enrolled.

Bruce Fuller, a K-12 education policy expert at the University of California at Berkeley who has been studying the New York program, called the progress “enormous” but said it falls short because there are an estimated 100,000 4-year-olds in the city.

And he questioned whether it makes sense for taxpayers to subsidize preschool for middle-income and affluent families.

If the idea behind free preschool is to prepare low-income children for elementary school and give them the same support wealthier children have, free preschool for every child — regardless of income — could inadvertently reinforce disparities between rich and poor, Fuller said.

Fuller added that de Blasio has “earned a B-plus on reducing the cost of pre-K for the true middle class but a D-minus on narrowing early gaps” between demographic groups.

The mayor defended his policy of offering free preschool to any family, regardless of income, at a cost to state and city taxpayers of $445 million for fiscal 2016.

“I don’t know of a middle-class family who doesn’t regard [pre-K] as fundamental,” de Blasio said, noting that his own children both attended full-day preschool. “And we saw extraordinary gains.”

“I’ve talked to plenty of middle- and upper-class parents who were struggling to find a good seat,” de Blasio said. “They appreciate not having to pay that extra $10,000 or $15,000 a year.”

The mayor said that there will always be parents who don’t want to send their 4-year-olds to school for various reasons but that he hopes that as the program matures and its reputation spreads, holdouts will decide to try it.