Detective Superintendent John Bassett, in charge of the inquiry, said that the motive for the attack had almost certainly been sexual, although it was not yet clear whether Rachel Nickell, 23, had been raped.
Miss Nickell had been walking her dog on the common in south-west London at about 10am on Wednesday. Her two-year-old son Alexander, who was bruised and scratched, was found clinging to her body in a secluded copse. According to some reports, a man was seen washing blood from his hands in a nearby stream.
Last night, no charges had been brought against the man in custody, who is aged between 20 and 30. Police disclosed that he had been arrested at about noon on Wednesday on a housing estate in Putney, near the common, following a tip-off from the public. They obtained a warrant of extension yesterday enabling them to hold him until this afternoon.
Det Supt Bassett gave few details about the man. Mary Cunningham, who lives on the estate, said that he was a regular visitor to one of the flats.
Police are still searching for the murder weapon, believed to be a sheath knife with a five-inch blade and a brass hilt.
Miss Nickell was a part-time model who lived with Andre Hanscombe, Alexander's father, in Balham, south-west London. Her body was identified yesterday by Mr Hanscombe, 29, a motorcycle courier. An inquest which opened at Westminster Coroner's Court was told that she died of multiple stab wounds.
Det Supt Bassett said that police had not talked to Alexander, who was still in shock, but planned to speak to child psychologists today about the chances of interviewing him. He is with his father at the home of a relative.
Mr Hanscombe said that Alexander had not spoken since the murder. 'I don't know how he will be in the future, but they tell me he is small enough that he won't remember much.'
He said that the killer must be known to somebody. 'No matter how they feel about this person, please come forward before he destroys somebody else's life.'