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Two points separate the Premier League's top three clubs after leaders Chelsea were held to a 1-1 draw with Manchester United on Sunday.
Elsewhere, Arsenal continued their top-four push with a 2-0 win over bottom side Newcastle, Aston Villa made it back-to-back wins under manager Steven Gerrard with a 2-1 victory at Crystal Palace and Leicester City won 4-2 in an entertaining match against Watford.
Brentford ended their five-match winless run with a 1-0 victory at home against Everton, there were goalless draws between Norwich and Wolves, and Brighton and Leeds, while Tottenham's match at Burnley was postponed because of heavy snow.
Check out my team of the week and then make your own selections towards the bottom of the article.
Goalkeeper: David de Gea (Man Utd)
Two saves in the first three minutes by David de Gea made it clear what sort of afternoon Manchester United were in for.
United got to half-time without conceding and that was down to De Gea. The same was true on Tuesday night in Spain against Villareal in the Champions League. Once Jorginho made that monumental mistake it gave United the initiative.
Why Jorginho didn't simply head the ball back where it came from, goodness only knows. I just hope that when De Gea has one of his dodgy periods (and he will) his team-mates provide him with the support he has shown them of late.
Did you know? Chelsea are now winless in their last eight Premier League games against Manchester United. They endured a longer run without a league win against United once before - a run of 10 games between 1938 and 1950.
Defenders: Antonio Rudiger (Chelsea), Virgil van Dijk (Liverpool), Nuno Tavares (Arsenal)
Antonio Rudiger: He made a fabulous tackle in the first half which set the tone for his entire contribution to the match. Rudiger loves a battle and the fixtures against Manchester United, Liverpool and Manchester City are made for the likes of him.
There are times, of course, when Rudiger is actually looking for confrontations even when they aren't there - in which case he will create his own. He's not averse to scoring the odd goal every now and then either, provided it's on his head and he's attacking the opposition's goal.
He was presented with a glorious chance during the closing minutes of the game against Manchester United and blasted the ball high, wide and handsome. Virgil van Dijk had a similar chance against Southampton and buried it. Two world class defenders and both make my team of the week.
Did you know? In their last two Premier League matches at Stamford Bridge, against Burnley and Manchester United, Chelsea have attempted 49 shots and had 100 touches in their opponent's penalty area, but failed to win either.
Virgil van Dijk: There can't be many central defenders in the world who could have scored a goal like Van Dijk did against Southampton.
To strike the ball on the volley and direct it towards goal with enough pace to beat the keeper from just inside the box was impressive. This was a walk in the park for Liverpool.
Meanwhile Southampton, without Danny Ings up front and Jannik Vestergaard at the back, look mediocre. If they don't address these two positions in January the Saints won't go marching on but marching out of the league.
Did you know? Six of Van Dijk's 11 Premier League goals for Liverpool have been assisted by Trent Alexander-Arnold.
Nuno Tavares: This kid had passed me by but he's certainly on my radar now. I've always admired fullbacks who are not afraid to try and have an impact on the game.
If Tavares wasn't running into the opposition's half with the ball he was trying to put one of his team-mates in on goal. The assist for Bukayo Saka's goal was quite brilliant.
As for Newcastle, it's clear Eddie Howe has got an awful lot to do. However, I couldn't help but notice that the new manager has invested some faith in Jonjo Shelvey, who had a decent game. If Howe can get Shelvey fit and focused for any length of time, Newcastle might save themselves quite a bit of money in the transfer market come January.
Did you know? Arsenal have never lost a Premier League home game against the side starting the day bottom of the table.
Midfielders: Bukayo Saka (Arsenal), Thiago Alcantara (Liverpool), James Maddison (Leicester City), John McGinn (Aston Villa)
Bukayo Saka: From the moment the game kicked off Saka looked like a player who meant business.
He gave Newcastle a bit of a fright in the game's early exchanges, but the Geordies survived. However it was only a matter of time before this brilliant young talent exposed Newcastle frailties down their right flank.
Saka scored a fabulous goal matched by a super display of commitment and effort. I'm looking forward to watching this lad take his next penalty, just to see how much he's grown since the European Championships.
Did you know? Saka has scored or assisted in each of his past three Premier League games against Newcastle.
Thiago Alcantara: The Spaniard has never totally convinced me that he has acclimatised to the intensity of the Premier League.
There is no doubt he can play, but when the game becomes physical or hectic I often feel his lack of pace limits his ability to engage and the game starts to pass him by. Liverpool's dominance of Southampton allowed them to control the game - Thiago in particular.
He's masterful on the ball and has the most velvet touch. This is all very well when your team is in control but Liverpool need the occasional rottweiler (enforcer) in midfield and that, he is not.
Did you know? Liverpool have scored 39 goals in 13 Premier League games this season, their highest goal total at this stage of a top-flight campaign.
James Maddison: A goal by Maddison - and he set up a couple as well in appalling conditions - gave Leicester the impetus they needed to take the game beyond the reach of Watford.
It was the best game I've seen Maddison have for some time. It was just as well because the Hornets, rejuvenated under the watchful eye of Claudio Ranieri, had a real go.
Watford certainly know how to score goals but if Ranieri could teach his defenders how to defend then they might be a team we start taking seriously.
Did you know? Maddison scored and assisted in a Premier League game for the first time on Sunday, in what was his 111th appearance in the competition.
John McGinn: It might be two wins out of two for Steven Gerrard but it was the performance by McGinn that clinched the victory at Selhurst Park for Villa.
McGinn was unlucky not to have scored earlier in the match when he cut in from the right and bent a left-footed shot just past the far post with Vicente Guaita beaten.
This was a disappointing display by Palace who had been playing well recently. With Michael Olise back, Christian Benteke among the goals again and Wilfried Zaha looking menacing, it looks like Patrick Vieira might be on the verge of creating a really decent team in south-east London.
Did you know? Aston Villa's McGinn has equalled his best goal tally in a Premier League season in just 12 appearances this campaign. He scored three in both 2019-20 and 2020-21.
Forwards: Mohamed Salah (Liverpool), Jamie Vardy (Leicester City), Sadio Mane (Liverpool)
Mohamed Salah: Some of my readers will be a little surprised that Diogo Jota is nowhere to be seen among my selections this week. That's because I thought his performance (two goals or not) was ordinary in comparison to Sadio Mane and especially Salah.
The Egypt international is a joy to watch at the moment. There isn't a better striker in the word than Salah when he is playing for the team and not merely chasing goals. Against the Saints he was superb.
What amazes me about Salah is his strength. He is lightning quick and extremely strong on the ball. However, Liverpool have a problem. They lose both Mane and Salah to the African Nations Cup in January. I wonder what impact that is going to have on their title chances.
Did you know? Salah has been directly involved in 25 goals in all competitions this season, with 17 goals and eight assists - more than double that of any other Premier League player.
Jamie Vardy: Vardy was having another one of those parties again.
The former England international hasn't made my team of the week for some time now but such was his performance against Watford the centre-forward put me in an impossible position.
Vardy, to his credit, plays the game with such passion and enthusiasm it's infectious. His link-up play with Maddison was a bit special against Watford and it's nice to see good players playing well at the same time.
Did you know? Vardy has equalled Ian Wright's record of scoring 93 Premier League goals aged 30 or older. Only Harry Kane (108) and Mohamed Salah (106) have more goals in the competition than Vardy since his first game in January 2017.
Sadio Mane: This was Mane at his best. The Senegal superstar was everywhere against a Southampton side that never came to terms with the demands of Anfield.
When Mane is at his best he is not just scoring goals but affecting the play. He was desperately unlucky to have a brilliantly headed goal ruled offside after being on the end of an awful challenge by Jan Bednarek, who was lucky to remain on the pitch.
Last week I criticised Jurgen Klopp for getting involved with Mikel Arteta and inflaming the crowd over a challenge that was tame in comparison to Bednarek's tackle on Mane. Fair play to Klopp, he didn't try to influence the referee's decision in any way but Bednarek should have gone.
Did you know? Liverpool have now scored two or more goals in 17 consecutive games in all competitions. They are only the second English top-flight club to achieve that, after Sunderland in 1927.
Pick your XI from our list and share with your friends.
The Crooks of the Matter
The humiliating departure of Ole Gunnar Solskjaer from Manchester United begs the question whether he should have been given the job in the first place.
The board, having unceremoniously booted out a United legend, have once again put their head on the chopping block with the fans who insist they appointed a manager worthy of their club and the players they demand.
They have gone for Ralf Rangnick. Ralf Who? I can almost feel the disapproval and contempt for the appointment buzzing around Old Trafford already, and that's before the new interim coach has taken up his position.
The problem with teams like Real Madrid, Barcelona and Manchester United is they are a victims of their history as much as they are their success. What's more, there can only be a handful of managers in the world with reputations big enough to manage those clubs - let alone keep control of those dressing rooms.
I totally understand why Zinedine Zidane would turn the job down. Why would he come out of retirement and put his credentials on the line having achieved the impossible and won three Champions League titles with Real Madrid?
There is simply too much to do at United. Neither do I see any sense in Mauricio Pochettino leaving PSG. If he can't win the Champions League with Lionel Messi, Kylian Mbappe and Neymar he can certainly forget winning it with United.
So who? The reality is United have boxed themselves into a monumental corner. There were managers available but they have dithered and procrastinated and paid the price for both.
They are now faced with the unenviable task of taking another risk or, better still, trying to find another Alex Ferguson. Whichever it is, I don't envy them one little bit.
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