"Supporters attending matches this weekend are reminded to be vigilant at all times and stay alert, not alarmed".
He urged anyone who had seen what happened, or had taken photos or videos of the bombing, to come forward. "For strong investigative reasons we will not give any more details on the man we arrested at this stage".
Hounslow is about six kilometres from Sunbury, where police raided and searched a building in connection with the bombing.
Turquie - Galatasaray : Bafetimbi Gomis brille encore
De quoi porter son total de buts à 7 en 5 matches et permettre au Galatasaray de conforter sa place de leader du championnat.
British counter-terror police are being assisted by MI5 and are scouring CCTV in their hunt for the bomber.
Witnesses described seeing a fireball filling the carriage, with passengers suffering burns, as well as other minor injuries during the rush to escape.
May said the military would be providing support to police and would replace officers on guard duty at national infrastructure sites that are not accessible to the public.
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US President Donald Trump denounced the attack on Twitter saying: "Another attack in London by a loser terrorist. I just had this feeling, 'Was I going to die?'" she said.