The Republic of Ireland have put their heart and soul into this World Cup qualification campaign, but when it mattered most they were short of the one thing that really matters – quality.

That was served up on a silver service for Denmark by Tottenham Hotspur’s Christian Eriksen, who scored a magnificent hat-trick, stroking two shots into the net like he was knocking balls into an empty goal at the end of a training session, before smashing in a third.

Ireland were forced to bend the knee in the presence of his brilliance. Denmark had one world-class player, Ireland had none. It can make small margins look like a gulf in class.

Ireland were crushed, but the suggestion they had been humiliated at the end of a campaign in which they were fourth seeds in their group and still finished second, drew an angry response from manager Martin O’Neill, who walked out of a post-match television interview.

“We were well beaten, no question about that,” said O’Neill. “We were made to pay for some sloppy defending, two really poor goals to concede and when we tried to chase the game, we were punished.

“But we were beaten in a play-off. We have fought tooth and nail to get here, so the sense of disappointment is huge. But we have played 24 competitive games and lost four. This is one of them.

“We were beaten by a side that was better than us and has a world-class player in the team, who was absolutely magnificent. Eriksen is top class and he showed it.”

Eriksen now belongs in an elite group of players in the modern era, alongside Gareth Bale, Cristiano Ronaldo and Zlatan Ibrahimovic, who have carried their national teams to new heights, defining the big games with the force of their ­individual talent.

In the end, Ireland gambled and lost. O’Neill tried something different and it did not work. Having smothered Denmark in the first leg in Copenhagen, he tried to attack them. He might regret it now, but in truth, nothing was likely to stand in Eriksen’s way. He and Denmark fully deserve to be at the World Cup next year.

By the time Ireland play in one, it will have been at least 20 years since their last appearance. An entire generation will have grown up without seeing their country play in the biggest tournament of all.

Then again, for all of the sense of entitlement in Irish football, the strange assumption they should be playing on the biggest stage, World Cups tend to take place without them. They have only made three, the last of which was in 2002, when they had a squad packed with ­Premier League players. They do not have that luxury anymore, but this was still a huge let down, the sort of anti-climax that turns a party into a wake. They have felt like this before. This was the fourth World Cup play-off they have lost, but this had been their chance to erase those bitter memories to the point of deletion.

It had the perfect opening paragraph, Shane Duffy heading Ireland in front from Robbie Brady’s free-kick. Ireland had other chances, a flick from Daryl Murphy almost turned in Cyrus Christie’s cross before James McClean, after a lovely move down the left, was played in behind by Robbie Brady. But, from a tricky angle, he dragged his effort wide of the far post.

Minutes later, Denmark were level. Harry Arter was beaten too easily from a short corner by Pione Sisto and, although Andreas Christensen sliced his shot against a post, it rebounded in off Christie.

A setback turned into a disaster when Stephen Ward was robbed by Yussuf Poulsen. The ball was moved to Eriksen and Denmark’s star player found the net via the underside of the crossbar.

Needing two goals in the second half, O’Neill went for broke and ended up broken. Off went Arter and David Meyler – on came Aiden McGeady and Wes Hoolahan. Caution was no longer a viable policy, but he was sending on two attacking players from the Championship, not the Premier League.

Eriksen made full use of the space, curling in a second goal and punishing some poor control from Ward to grab a third. Nicklas Bendtner added a fifth with a penalty.

“Eriksen is one of the best players in Europe,” said Denmark coach Age Hareide. “We have seen that against Real Madrid in the Champions League and tonight. He is in the top 10 players in the world, he has this ability to score goals, to make assists and find space, he is a special player. As soon as Ireland gave him space, he hurt them.”