NEW YORK — Sloane Stephens probably will be able to beat Serena Williams again someday.
Just not Sunday.
Williams, defending U.S. Open champion and the top-ranked player in the world, appears well on her way to winning a 17th Grand Slam title after she dominated 15th-seeded Stephens, a 20-year-old who is considered, in the United States anyway, as the heir to the top tier of the rankings where Williams and her older sister Venus have mostly stayed for over a decade.
Williams defeated Stephens, 6-4, 6-1, for a fourth-round victory in Arthur Ashe Stadium in front of about 20,000 fans.
While the score appeared lopsided, the tennis wasn't.
Both players hit high-quality shots. Williams commanded the net, lunging for volleys, and Stephens hit some forehands clocked at over 100 mph that would skip off a sideline or baseline.
Williams executed two lobs in windy conditions where the ball would hurry and scurry in unplanned directions, and Stephens twice passed a net-bound Williams, once with a curving backhand and once with a blasted forehand.
In the closer first set, Williams, 31, earned the first service break and a 4-2 lead with a forehand winner, but Stephens stared her down by getting that break back immediately when Williams, uncharacteristically, served back-to-back double faults, a little hint that she was worried.
But in the final game of the 52-minute set — a game that lasted 14 points and in which Stephens saved three set points — Williams earned the crucial service break by moving Stephens around the court with powerful ground strokes until Stephens slapped a forehand into the net.
Williams got the decisive break in the fourth game of the second set with a massive forehand return winner that elicited a shout of "Yes!" And then Williams held at love for a 4-1 lead and the fight seemed to leave Stephens.
The two exchanged a hug at the net and Williams put her arm around Stephens' shoulder.
Williams agreed about the level of play.
"I definitely think it was a high-quality match," she said. "I don't think either of us had an overwhelming amount of unforced errors. We both came out today to play."
"I thought I played good," Stephens said. "There were times I played really good tennis. I played pretty solid. But, I mean, obviously she's No. 1 in the world for a reason. I thought she played really well herself."
Williams had 22 winners to 13 unforced errors, while Stephens had 15 winners and 29 unforced errors. Neither player was cautious. Both were willing to aim for lines and hit as hard as they could.
Stephens had upset Williams in the season's first Grand Slam event, the Australian Open, and marked herself as a big-game player.
Next up for Williams, in the quarterfinals, is crafty Spaniard Carla Suarez Navarro, who upset eighth-seeded Angelique Kerber of Germany, 4-6, 6-3, 7-6 (3).
Stephens said she "embraced" the idea that she might be considered the heir to Williams' top spot.
"Right now I'm carrying the little torch and I'm OK with that," she said. "I embrace it for now."
It is a torch Williams plans to hold onto a little longer.
While she said she didn't play Sunday's match with revenge on her mind, she went out with the intention of fighting until the end.
"When I'm playing . . . I'm not thinking about being a star. I just want to play tennis and I want to do really good at it still. It's not about the stage for me or the ranking or the title, it's just about getting that ball in."
Defending men's champion Andy Murray was businesslike in his dispatching of Germany's Florian Mayer, 7-6 (2), 6-2, 6-2, in a third-round match.
Murray may not have come through unscathed, though. He put an ice pack on his back to cool off and in doing so he said he felt a painful pull in his left forearm.
Sunday night, Agnieszka Radwanska, the third-seeded woman, was upset by 24th-seeded Ekaterina Makarova of Russia, 6-4, 6-4, the fourth time since 2007 that Radwanska left Flushing Meadows in the fourth round. Li Na, seeded fifth, rolled past Jelena Jankovic, 6-3, 6-0.
Novak Djokovic, No. 1 on the men's side, routed Joao Sousa, 6-0, 6-2, 6-2.
And 25-year-old wild card Tim Smyczek of Milwaukee, the last U.S. men's singles hope, lost to Spain's Marcel Granollers, 6-4, 4-6, 0-6, 6-3, 7-5. For the first time in history, no American man advanced to the fourth round in singles at the Open.
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