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It was a triumphant return to Spurs for Mauricio Pochettino who got the biggest win of his short Chelsea tenure.
City thoroughly dominated shell-shocked United in the second half.
Victory pushed Arsenal up to second place with 24 points from 10 games, two behind Tottenham Hotspur.
Arsenal came into the match having not beaten City in the league since 2015, a run of 15 games -- their longest winless streak against a single opponent in league history.
Manchester United's campaign of woe continued with a 1-0 home loss to Crystal Palace but there were no such troubles for Arsenal who, along with Spurs, closed the gap to leaders City to one point with a rampant 4-0 win at Bournemouth.
Liverpool won 3-1 while Arsenal were held 2-2 by Spurs.
Chelsea lost at home to Nottingham Forest in the Premier League.
Liverpool snatched a 2-1 win after going a goal and a man behind in the first half.
Mikel Arteta's side were held 2-2 by Fulham leaving the Gunners with seven points.
West Ham won 3-1 afte Enzo Fernandez missed a penalty to send Chelsea 2-1 up.