May said the developments meant the Joint Terrorism Analysis Centre (JTAC), the independent body which sets the threat level, had decided it should be lowered from its highest rating "critical", meaning an attack could be imminent, to "severe".
The statement from Greater Manchester Police and United Kingdom counter-terrorism police said they had a 1,000-strong team working "around the clock" on the probe. "The country should remain vigilant". "There will be more arrests". Police used an explosive device to get into a property to make the arrests.
British officials have confirmed Adebi had recently returned from Libya and the officers said police needed information about his movements from May 18 when he returned to Britain.
Greater Manchester Police (GMP) said a 19-year-old man had become the latest suspect detained over Monday's atrocity, which left 22 dead.
Earlier on Sunday, Home Secretary Amber Rudd said investigators can not be entirely sure that Salman Abedi's terror network has been dismantled - and warned other members of the bomber's group are "potentially" still at large.
Some hours later, police cordoned off a large area in the Moss Side area of south Manchester and houses were evacuated with a bomb disposal unit sent to the scene.
Another place searched was an apartment in a Manchester high-rise that British media say was rented by Abedi in the months before the attack.
"These searches are connected to Monday's attack on the Manchester Arena, but this is a fast, ongoing investigation and we are keeping an open mind at this stage".
Ariana Grande has called off two London concerts and five in Europe after a suicide bomber killed 22 people at her performance in Manchester, the United States of America pop singer's representatives says.
Earlier this week a source with knowledge of the matter told Reuters the security services were managing 500 active operations involving some 3,000 people thought to pose a threat.
There are a number of high-profile events over the weekend including soccer cup finals in London and Glasgow, and the Great Manchester Run.
Police also released more details about the bomber's actions just before he detonated his device at Manchester Arena.
And extra security measures were in place for Saturday evening's gig by the Courteeners, an indie rock group, at Old Trafford cricket ground - where 50,000 music fans attended the city's first major gig since the attack.
NEW YORK-Pop star Ariana Grande promised to return to Manchester to play a charity concert following a suicide attack at her show, as she urged fans to respond to the tragedy with love.
"We can't directly link these to the events of Monday night and are continuing to monitor the situation", he said.
Polls suggested the lead of May's ruling Conservatives over the opposition Labour Party has narrowed since the Manchester attack but she is still on course to win comfortably.
May said Labour leader Jeremy Corbyn was saying Britain was to blame for the bombing.
"I've always had a pride in this city, ever since I was born", he said.
A poll on Thursday put May's Conservatives five points ahead of Labour suggesting a far tighter race than previously anticipated.