Chants were directed at Arsenal throughout the game, while some home fans even celebrated the goals from Erling Haaland that have put Manchester City on the verge of yet another Premier League title.
Asked about the chants aimed at Arsenal, Spurs boss Postecoglou said: 'Not interested mate. I don't care. I'm just not interested mate. Maybe I'm out of step but I just don't care.
'I just want to win and be successful at this football club, that's why I was brought in. How other people feel and what other people's priorities are is zero interest to me.
'I know what's important - building a winning team, and that's what I need to concentrate on.'
Tottenham frustrated title-chasing Manchester City for 50 minutes but Haaland broke the deadlock with a tap-in and then put the game to bed with an injury-time penalty.
Haaland and City's second came shortly after Son Heung-min had a glorious chance to equalise only to be denied by Stefan Ortega, who replaced Man City No. 1 Ederson in the second half due to a head injury.
On Spurs' performance against the champions-elect, Postecoglou added: 'It was a tight game and in the big moments we weren't able to capitalise and they were.
'I thought we were in the game the whole time, even at 1-0 we had a great chance to go level.
'In the second half we were chasing the game so they got a little more space but I thought we were well in it up until that point [the second goal]."
'We lost the game so it was a missed chance for three points. We had the opportunities and we had to punish them.'
Tottenham midfielder James Maddison, meanwhile, said: 'Any result at home where you don't win is disappointing.
'I am not naive enough to think we were not playing one of the best teams in the world because we were. Always disappointing to lose at home in front of our own fans.
'We were in the game for large parts, took the game to City and played some really good football but ultimately it results in zero points.'
Tottenham visit bottom-placed Sheffield United on the final day and need a point to guarantee a fifth-place finish ahead of in-form Newcastle and Chelsea.
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