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If you have large areas like here in the white upper left corner,
why can't black just put a stone in upper left corner? Then white would lose one point. Or white would have to capture it and get a point instead, but then there would be three white stones within the area and that would mean again for white to lose three points.
So is it just convention to end the game here, when black won't be able to create a living group within the area?
Suppose there is a black stone in the corner but it is impossible for black to survive, no matter how well played.
By convention, both sides recognize this and the black stone is counted as a prisoner.
If either side doesn't accept the convention, both sides will alternately play until the situation is resolved (in favour of white).
At that point, black and white will have played the same number of stones each. White loses a point for each one it played (loss of territory), and white gains a point for each one that black played (taken as prisoner). There is no net change in the score as compared with accepting the initial situation and neither side playing any more.
Note that the above is based on territory scoring rules, where points are given only for unoccupied territory (and prisoners). Under area rules the non-prisoner stones on the board are also counted as part of the territory, so any stones played by white into its own territory would not cost white anything.
This depends on the rules used.
In rules using area scoring (Chinese style), a player's points include both the empty intersections they surround, and their stones. In this case, any extra moves made inside a player's own territory do not cost points, unless there are dame left, since the move could have been used to take one and with area scoring that would be worth points.
In some rules (AGA), a player passing must give their opponent a stone (a pass stone) which counts as a captured stone, and is worth one point if territory scoring is used. Therefore, for every move White makes to capture the Black stones inside White's territory, Black must also make a move (either inside their own territory, losing a point; or inside White's territory, eventually to be captured for a point) or pass (losing a point).
Note that AGA rules also allow for area scoring, in which case pass stones are not necessary.
See:
In traditional Japanese rules (which use territory scoring causing extra moves inside a player's own territory to cost points) there is a special confirmation phase after the game has (provisionally) ended. During this confirmation phase, a particular situation can be played out to determine if the contested stones are alive or dead, but the confirmation play is taken to take place "virtually", and does not actually modify the board. The phase of confirmation play also has special ko rules with somewhat odd results, like the infamous case of a bent-four in a corner being dead as a rule.
See:
If Black plays in that area, White could ignore it or pass and gain a point - the black stone would be agreed as dead, and White still gets the point it's sitting on. Black can only force White to surround and capture the stone if they're threatening to capture some of White's stones - for example if they cut off part of the wall, and it creates a capturing race. If Black keeps playing in that area, White has to decide when to start answering - if they wait too long, Black will make a living group.
Your final line has the correct question more or less answered. It is convention to end the game because black would not be able to create a living group. What is important is that both players agree to this. A lesser-skilled player (black) may not be convinced and request to play it out.
As the board sits, without visible captured (removed) stones, white wins by 0.5 komi (or higher) and would not want to play one extra stone. TimK's answer about waiting too long would apply and could turn the result.
It's extremely challenging (or impossible) to create life in a small corner as presented, hence the reference to the lesser-skilled player.
Simple answer:
If black did play in the upper-left, white does not necessarily have to capture that piece, as it's dead and captures no territory. So really, we are trying to capture as much territory, but that in-turn also means we don't waste pieces.
