With a maximum return of 18 points from six games, Liverpool are flying in the Premier League, but Adrian is not getting complacent.
Liverpool's stunning start to the season will count for nothing if they do not maintain their high standards, Adrian has warned.
The Spaniard continued to deputise for the injured Alisson as the Reds claimed a 2-1 win at Chelsea on Sunday, putting them five points clear at the Premier League summit.
But Adrian urged his side to keep pushing for more, knowing reigning champions Manchester City are ready to pounce on any slip-ups.
"We keep fighting and we keep that momentum going and keep it in a positive way, of course," the goalkeeper told the club's official website.
"We have done really well but we've done nothing yet so we need to keep going and keep fighting for the next one. We are on a good run in the Premier League with good energy and intensity.
"In the first half [against Chelsea] we did really well and I think we deserved the three points. Obviously they pushed in the second half but we defended really well."
While Jurgen Klopp's side boast a 100 per cent record in the top flight, the defence of their Champions League crown started with a 2-0 defeat at Napoli.
Adrian conceded it was a tough loss to take but he took comfort in their response at Stamford Bridge, where first-half goals from Trent Alexander-Arnold and Roberto Firmino were enough to get the job done.
"The other night in the Champions League against Napoli we lost," he said. "I don't think we deserved to have that result and it was really hard for us, but we recovered well. We are back in the Premier League and back to winning.
"It's always important to have a good connection between all of the players, also the fans, the crowd, the manager, the staff – everyone.
"We are like a rock now and we are together. We are enjoying this moment but we need to keep looking forward for more points."
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