First off, count the current score. If you're winning, do not invade. Just make sure you keep white small enough that you still win.
Let use further presume that black would be losing. You counted, so you know by how many points. A move around c could be parried with a sacrifice of two white stones while sealing off the rest of the upper right corner ...
Liam showed a good way to kill the invasion already, but you can be even more mean and play 2 at 4:
If white then plays 3, you just draw back instead of playing below 1, and you avoid a cut at 4:
I know this is a textbook situation, but I do not practice tsumego, so I had to read through this. It was quite tricky and took some time to, hopefully, get right.
The strongest (i.e. hardest to correctly answer) resistance I found is this:
$$ . X 3 5 2 . . . |
$$ . X O O 1 . 4 . |
$$ . X X X O O O . |
$$ . . . . X X O . |
$$ . . . . . ....
I decided to revisit this post with the help of Leela Zero.
LZ's assessment is that there is no good move for black above the 11th line. The sequences are fascinating, here is the start of one:
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . 6 . |
$$ | . . . . . . . O O ....
The black stone at Q18 is in a ladder, albeit a short one. If black pushes this, with R18, black will eventually be killed and white will live. R18 is a strong KO-threat, however, because if it is not answered black can kill the white group.
If black plays P29 white must capture at R18. If black pushes at O19 the capture is again forced and white will ...
I've seen and played a version of this were you put the black stones along the center lines of a regular 19x19 board so you carve out a 9x9 space. White is then supposed to create a living group inside. For players of equal strength this is approximately a fair game. I'm not sure what the result of perfect play for either side would be but I'm pretty sure ...
Given the size of White's territory, you are probably behind. Then the question is, can you catch up, and if so, how? I'll save the first question for last, and answer the others in order:
2) You have the choice of ONE invasion point, a, b, or c (or some alternatives mentioned by other posters). Choosing one strengthens White sufficiently to preclude the ...
Many question in one post ...
An invasion on a should work, because the corner is not yet 100% secure. White will probably build up strength to the outside leaving no potential on b. White will probably get the initiative afterwards.
An invasion on c looks very big, probably bigger than the corner invasion because of the two white stones to be captured. ...
In general, corner enclosure is a great thing. If you can manage to do so in the early game, you have a big advantage. You gain a lot of free moves on the outside, since the opponent must protect his corner. You will then build huge walls with a big impact and you can greatly develop to both sides and you have also impact to the center.
(But note that this ...