Example:

Stacks 100BB
Button open-raises A5s to 3BB. Only the big blind calls.

Flop AK6r, the big blind checks to the button. If the button c-bets A5s:
- better hands like AK-AT & 2-pair & sets will call
- worse hands like A4-A2, KJ-, QQ-77, 55-22 will fold

So in this specific case, does c-betting accomplish anything? (I ask because it seems many people seem to blindly c-bet any flop)

The simple answer to the title of the thread is YES. I check back about 40% of time when IP in cash games.

A5s is also a pretty standard check back on that flop. A c-betting range for that spot is usually very polarised, with something like AT being the worst top pair that should bet. (You can bet weaker aces if you pick a small size though. I usually go big on AK6r, and save my weaker Ax for delayed c-bets or use them as bluff-catchers).
People that "blindly c-bet any flop" tend to be losing players.

A2 will not fold, KJ will not fold and QQ-99 are almost never in his range. He will still fold 78s and stuff like that but not those hands.

C-betting also lowers the risk of your opponent going over the top and bluffing to try and represent a stronger A. It seems like you're saying its pointless to make a c-bet because if he calls hes got you beat anyways, but it just gives him an easiest shot to push you out of the hand.

If BB has 1010-JJ-QQ (not likely he flat called your raise preflop, but using it as an example) and you check the flop, its enabling him to become the aggressor and have you second guessing the strength of your ace.

This is exactly the reason to check back, to give him an opportunity to bluff turn or river. What you're trying to do is prevent him from ever bluffing by betting yourself which is going to incinerate money in the long run.

I'd like to play against you, because you're not going to push me off the hand. I check back the flop and call every turn bet and most river bets too. By checking back the flop, I can pick up 2 bets from a bluffer, when c-betting will usually just make him fold and give me no value at all all.

I know a lot of beginners don't understand this, because all they know is "You should continuation bet often", but I had a lightbulb moment when I realised that bluff-catching is every bit as important as value-betting. You can't bluffcatch if you don't check back your bluffcatchers.

That’s factually incorrect.

Cbetting is the he only way for your opponent ‘to come over the top’. By betting after his check, you reopen the action for him and he has the option to raise. If you check behind, the turn card will come out instead.

I think what he meant was that if you check the flop he can bet the turn, which of course is the desired result when checking back top pair.

Yes, c-betting accomplishes some things on this board. You can fold out hands that have outs against you, like JT, or 88 and get some value from Kx. You could also induce a bluff raise, although that might be a bit of a mixed blessing as we might very well fold to it.

But I do think we should frequently check back with the initiative, even in a heads up pot but especially multiway.

This flop texture in particular is very conducive to more passive WAWB lines in general, as it is a very dry/lockdown board, and to checking back as the PFR in specific, as it is one where you should get a lot of credit.

but doesn't checking give his kxs combos a free opportunity to become two pair or trips? I spose you both get that same opportunity so it's neutralised.

question.. if check/check on flop, villain bet and you call on turn, villain check on river, should you check as well or bet for value say you didn't get 2 pairs?

Depends on the runout, but you should bet a good amount of the time.

+1 To Duncelanas. On some runouts A5 will be high enough in your range to go for 2 streets (by calling turn and betting river), but on others checking back the river w/ top pair will be safer, especially if you have other combos that are, or made, stronger hands. In game, I often value-bet small on the river if villain "probed" the turn. He'll often sigh-call with Kx or 3rd pair.

+1

Play your enemy - not just your cards. (hyper)LAGs will gladly grab rope to hang themselves once you feign the tiniest sign of weakness.
The added bonus is: They made the mistake when they are actively deciding to bluff instead of passively calling an inferior hand to value bets. Losing on an own mistake (especially if it's a two barrel or three barrel bluff) is way more tilt-inducing.

Generally against loose passive Villains you can just value bet spots like this. Against more aggro Villains or just better players in general you want to check back hands like this occasionally.

This helps protect your checking range. In other words, if you check back the flop villain won't be able to just bluff away with impunity because you can still have some strong hands in your range.

this hand is all player dependent. some players its a must to bet. like those that arent going to play on anyway but will with a free card improvement to beat you.

dont bet against players that will bluff or value bet at you with a weaker hand. etc. etc.

here is a hand i played today. 10/20 blinds. lady gets in the game and like a fool shows she is a seasoned player by aggressive shuffling chips expertly. now i know how to play her.

soon a hand comes up her sb me bb. she opens for 50 i make it 150 she calls.
flop ace dd. she checks i check back. i know she will bet next card by her shuffling aggressively.

4th i make the nut flush she bets 400. i just call as she is drawing dead most likely.

river blank she checks. i bet 900 keeping it under 1000 and she calls. says something like she is glad another diamond didnt come. she must of had king ,queen with the king diamonds.
good pot with no chance of losing by checking back on the flop. and giving her the idea i might have nothing

This is the button vs. the big blind and in case you have read some latest books, you would likely be supporting the idea of betting with a top pair if there are many hands in the opponent's range that will call, like is the case here. Not sure if there is enough getting moved off the hand risk here. Betting all three streets is unlikely on the menu, and maybe not too much free card risk, so checking back is fine as long as you also check back enough junk.