Premier League leaders Manchester City showed the spirit of champions as they fought back from a goal down to beat West Ham. Eden Hazard dazzled for Chelsea and Sam Allardyce made a winning start with Everton.

Here are five things we learned from the Premier League this weekend:

City pass character test

For all the plaudits thrown Manchester City’s way for the sublime football they have played this season, there is a defining characteristic to the leaders that boss Pep Guardiola admires more than all the purist-pleasing play – grit. Flamboyant skill and lethal finishing have stolen the spotlight, but Guardiola is more heartened by City’s refusal to be subdued by opponents hell-bent on frustrating them, and never was that more apparent than in their 2-1 win over West Ham on Sunday. Falling behind on the stroke of half-time after struggling to penetrate West Ham’s massed defence, City simply imposed their will on the visitors in the second half, pressing them into submission until goals from Nicolas Otamendi and David Silva sealed their record-equalling 13th successive league win. It was the fourth straight decided by a late winner for City and Guardiola, whose side are eight points clear at the top, said: “You see how they run and how they fight, that’s why we are so proud.”

De Gea flips the script

As the full-time whistle sounded on Manchester United’s 3-1 win at the Emirates Stadium, United manager Jose Mourinho clenched his fist in victory while Arsenal boss Arsene Wenger trudged solemnly down the tunnel. But for a brilliant display from United goalkeeper David De Gea, the end-of-game reactions from those two old foes would have been completely different. Spaniard De Gea denied Arsenal on several occasions with a series of breath-taking saves, the best of which came minutes into the second half when he denied Alexandre Lacazette from close range before spectacularly saving on the rebound from Alexis Sanchez. United’s win kept them in the title hunt, so it’s little wonder that afterwards Mourinho hailed De Gea as the world’s best keeper.

Everton: Crisis, what crisis?

Sam Allardyce is used to walking into clubs in crisis and performing heroic rescue acts but the new Everton boss finds the Merseyside club in the top half of the Premier League table after their first win of his reign on Saturday. The survival specialist, who has saved Sunderland and Crystal Palace from doom in recent years, was in the crowd for the 4-0 win against West Ham in midweek but the 2-0 victory against Huddersfield was Everton’s first since he was confirmed in the Toffee’s hot-seat. Before hammering the Hammers Everton were 17th in the Premier League, just two points above the relegation zone. Now they’re tenth and eight points clear of the drop zone. “Getting a win in your first game – let’s hope it’s onwards and upwards, boys,” said delighted Allardyce.

Free-scoring Liverpool

Liverpool have had a mixed start to the season, with their shaky defending causing concern, but their recent form has been impressive and their attacking options are dazzling. Jurgen Klopp’s team, boasting the fluid talents of Roberto Firmino, Philippe Coutinho, Mohamed Salah and Sadio Mane, have scored 12 goals in their past three Premier League away matches after Saturday’s 5-1 win at Brighton and 25 goals in their past eight games in all competitions. They are now third in the league goalscoring charts behind the two Manchester clubs. “All of the forwards are really important for us. It’s cool to have these options,” said Klopp after his team climbed into to fourth with the win on the south coast.

Hazard dazzles

Chelsea’s Eden Hazard struggled with injury in the season’s opening weeks but now the Belgian is back to his best and terrorising Premier League defenders. Saturday’s brace against Newcastle took him to eight Chelsea goals in his past 10 matches in all competitions, and Blues boss Antonio Conte said he was looking forward to seeing his playmaker in action at next year’s World Cup in Russia. “It will be special for me to see England play Belgium in the World Cup, but I have to ask Gary Cahill and Danny Drinkwater not to kick Eden in that game,” he joked.