"The family is trying to have patience and faith through all of this," said Benjamin Crump, lawyer for the family of the slain teen Trayvon Martin.
"We want a very public trial so the evidence can come out and show people that the justice system works for everybody," he told CNN.
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"This was their son. It's now 42 days. They want an arrest. Just simple justice. They're not asking for anything that anybody else wouldn't want if this was their child."
State Attorney Angela Corey said she would not hold a special closed-door hearing to determine whether to bring charges into Martin's death, which had been set for Tuesday, but that the investigation was still continuing.
Martin was fatally shot in late February by neighborhood watch guard George Zimmerman, who was patrolling a gated community in the Florida town of Sanford.
"State Attorney Angela Corey has decided not to use a grand jury in the Trayvon Martin shooting death investigation," Corey's office said in a statement.
But it warned the decision "should not be considered a factor in the final determination of the case."
"At this time the investigation continues," it added, as prosecutors work to see if there is enough evidence to bring charges against Zimmerman.
Zimmerman, who has not been arrested or charged in the case, has said he acted in self-defense after Martin punched him on the nose, knocked him down and slammed his head into the ground.
A controversial Florida law allows the use of deadly force when a person senses a reasonable fear of death or serious injury, which Zimmerman says was the case when he shot Martin.
But the case has unleashed a national uproar over race relations and the right to self-defense in the United States.
Martin's family, who have launched an online petition seeking justice for their son now signed by some 2.5 million people, have urged an "imminent" arrest.
"Let's try to have faith in the system... because we expect an arrest to be made, that it will show that there's equal justice everywhere in America," they told CNN.

