Strong heat after flopped set

To me betting very small and folding vs a raise seems like a decent option. You could also check but then if villain bets you probably have to check fold.

I actually bet about half the pot again, which was a mistake. He turned over A9o. I felt that was actually ridiculously loose calling by him on every street.

Preflop was a loose call indeed. But OTF he can't fold. Turn bricks and you bet half pot so he pretty much has to call one more. OTR he has almost always the nuts so if he doesn't raise he should at least call always.

Interesting! I figured that he should be putting me on AT+, AA or TT, so he’s behind me with any of those hands. It’s unlikely I’m raising pre-flop and then firing two barrels with worse than that.

If I’m on AT, his only outs OTT are two 2s for split pot. If I’m on AJ, AQ or AK, then it’s those 2s and the three 9s. He’s dead if I’m on AA. With TT, only the two aces are outs. So if he’s assessing my likely hand, surely he realises he’s way behind?

Of course, you’re entirely right that he has to call or even raise the river (he called), because he’s only behind AT now and is almost certainly ahead.

Well I'm no expert in full ring games and I'm not sure if this is live or online game, but since you open from MP you likely have all AXs hands in your range. OTF you should really be betting other than Ax hands and sets. If you are not, you are very exploitable. Villain should just fold every time you bet if he doesn't have a decent hand and bet when you check because it means you are weak. So you wouldn't be balanced here. So it is reasonable that villain assumes you barrel some bluffs here too, as you should.

Then if we take a look at his minimum defending frequencies, he should defend about 60% of his range on the turn so you can't exploit him. He probably called the flop with quite many hands so any top pair should be fine as a defend here.

So, if villain puts you on that AT+,AA,TT range he should of course fold. But he doesn't have to put you on that range because your range should be wider. On the other hand he should probably consider that on this very dry board you can't be balanced and your range is too value heavy. So he probably shouldn't defend as much as would be theoretically correct. But can he still fold his top pair here? Maybe he could fold, but I think we can't say calling would be terrible.

Very detailed analysis! Thanks. The real ‘take home’ for me here is checking that river next time, because I’m almost always behind there.