There have been 18 world No 1s on the women’s tour since the autumn of 1975, when a computer replaced the abacus and common sense as the means of determining who was the best, the second-best and so on.

One recalls that it was not so long ago that the veteran correspondent of the Press Association was able to do the sums more reliably and more swiftly than the sport’s ready reckoner.

When the smoke has cleared from this year’s US Open, one of six of the present top ten will have taken the opportunity to claim the top spot, which is proudly being touted as a record number of prospective leaders but, in truth, goes to the heart of the weakness of the system and the much-of-a-muchness state of the women’s game. Anne Keothavong, the British No 1, rightly says that she has the talent to become a top-50 player, but so also do a legion of others.

Ana Ivanovic, Jelena Jankovic, Serena Williams, Svetlana Kuznetsova, Dinara Safina and Elena Dementieva are the super six and of those, only three have won a grand-slam tournament title: Ivanovic this year’s French Open, her first; Kuznetsova the US Open, since when she has gone 15 further of these tournaments without an inscription on a plate; and Williams a mighty eight, the last of which was the 2007 Australian Open when she committed late and demolished the field.

It could be argued that the picture has been clouded by the retirement of Justine Henin in May - when she was riding high on the Sony Ericsson Tour, even if her form had become a trifle shaky - and the shoulder injury that has rendered Maria Sharapova a less than potent force for the best part of six months. Is it not remarkable, too, that Venus Williams could add the US Open title to her Wimbledon triumph in July and not be in the reckoning?

On yesterday’s evidence, Ivanovic is not going to retain her position of No 1, which she has assumed twice. The first time came after her breakthrough triumph at Roland Garros and the second, two weeks ago, was when she was resting, having played a single tournament - in which she lost in the second round to Tamira Paszek, of Austria - since her third-round defeat at Wimbledon by Zheng Jie, of China. The Serbian clocked up 40 unforced errors in her 6-1, 4-6, 6-4 victory over Vera Dushevina, of Russia. It was a teeth-grinding experience.

Of five double faults, Dushevina served four in one game, the first of the second set, and yet still she won that set, drawing Ivanovic into a plethora of faulty strokes and sad looks. It was only when the compliment was returned, in the third set, that Ivanovic emerged with dignity upheld. It was not really an occasion to start blowing kisses, better to have disappeared sheepishly from view and hope that not many people chose to press the record button if they missed it.

Ivanovic has been injured; a dicky thumb prevented her from competing in the Olympic Games, so she flew to Australia in the interim to try to find answers why it would not properly heal. “I did MRIs and had lots of tests and X-rays, which found some cysts that were causing inflammation,” she said. “The doctor did lots of treatment and he is also a chiropractor, so he fixed my back and my neck and everything was so connected.”

As for Keothavong, after a number of years scraping results together, she is better as a player, more rounded as an individual and ranked No 87. “I am a lot calmer in myself, but if I feel the need to get into my opponent’s face, I’m not worried about that, either,” Keothavong said. “I am older and wiser, I feel more comfortable than ever.”

The proof of the pudding will need to be evident today against Francesca Schiavone, the No 25 seed from Italy.