Madison Keys has already been heralded by Serena Williams as the future No. 1 player in the world.

For now, Serena is the undisputed holder of that title as she pursues the Grand Slam here at the US Open. But the 20-year-old Keys took one step closer to another showdown with Serena when she knocked off Tereza Smitkova of the Czech Republic, 6-1, 6-2, in Arthur Ashe Stadium on Wednesday to advance to the third round.

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"It was amazing," Keys said. "I don't know if it's because of the roof now, but it didn't seem as big as I thought it was going to feel. It was amazing to play out there.

"I thought I played pretty well, so hopefully I can have many more matches on Ashe."

Keys will next face the winner between No. 15 Aga Radwanska and Magda Linette. She is 0-4 lifetime against Radwanska, but if she advances, Serena could be waiting in the fourth round. Williams won 7-6 (5), 6-3 over Kiki Bertens, Wednesday.

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"I think I have always kind of struggled with Radwanska's game a little bit," said Keys, who lives in Boca Raton, Florida. "She just seems to always kind of get that extra ball back. I think going into that match just a little bit more patient but still going for my shots."

Keys hired former world No. 1 Lindsay Davenport as her coach and then made a major breakthrough at the Australian Open in January when she reached the semifinals before losing to Serena in straight sets.

"It was an honor for me to play someone who will be No. 1 in the future," Williams said after the match.

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Keys followed that performance up with runs to the third round at the French Open and the quarterfinals at Wimbledon. On the year, she is now 26-14.

She was one of 12 American women who made it to the second round of the Open, and she and Madison Brengle then both advanced to the third round early Wednesday.

Brengle, a Delaware native, took out fellow American Anna Tatishvili, 6-3, 6-2.

"It's great stuff," Paul Annacone said on The Tennis Channel of the success of the Americans. "Between the men and the women, that talent pool is getting deeper. The deeper the pool the more likely you're going to have great players."

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Keys could be that next great American player in the post-Serena/Venus Era. But in order to get another shot at Serena, she must get past the next round.

"It's so special to be able to get as far as I've ever been at this Open," Keys said. "I'm excited to keep it going and try to get to the next round.

Follow Adam Zagoria on Twitter @AdamZagoria for updates throughout the U.S. Open.