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I have read somewhere, but unfortunately forgot where that the following enclosure is safe against the invasion at a.
$$ White invades at 'a' and supposedly dies.
$$ ---------
$$ -........
$$ -........
$$ -..a..X..
$$ -...X....
$$ -........
$$ -..X.....
$$ -........
What is the proper way to defend after white a? Is there a way for white to live, even in gote?
First of all, note that, unlike the more popular shape with a kosumi instead of a keima, this shape carries a lot more aji and is vulnerable to yose tesuji and approach moves.
$$ A lot less aji
$$ ---------
$$ -........
$$ -........
$$ -.....X..
$$ -...X....
$$ -..X.....
$$ -........
$$ -........
Oh, and by the way, an invasion is supposed to live in gote. Invading and living in sente is something you definitely don't want your opponent to be able to. In a shape with 3 (!) black stones vs a single white stone, living has to be super hard, and even a ko is a success for white.
The shape you showed, as is, is indeed save against an invasion. It involves a tesuji you need to remember though, and you also need to be able to read tsumego at, I guess, strong kyu level to be able to kill.
$$Wcm1 The invasion
$$ ---------
$$ -........
$$ -....2...
$$ -..1..X..
$$ -...X....
$$ -........
$$ -..X.....
$$ -........
B2 is tesuji to kill.
$$Wcm1 White can't live
$$ ---------
$$ -.98.....
$$ -.5..X...
$$ -0.O..X..
$$ -.7.X....
$$ -6134....
$$ -.2X.....
$$ -........
Why can't black simply block?
$$Wcm1 Black fails - white gets a ko
$$ ---------
$$ -..98....
$$ -.7.56...
$$ -..12.X..
$$ -..3X....
$$ -..4.....
$$ -..X.....
$$ -........
How is this shape dependent on the outside? Well, first, any approach move is virtually sente:
$$Wcm1 If black tenukies, the invasion becomes a huge danger again. But answering locally likely leads to overconcentration.
$$ ----------
$$ -.........
$$ -.........
$$ -.....X.1.
$$ -...X.....
$$ -.........
$$ -..X......
$$ -.........
Also, even nearby stones affect the tsumego severly:
$$Wcm1 Thanks to nearby stones, the fight is tough for black, he should not have tried to kill white in the corner.
$$ ------------
$$ -.OX........
$$ -XO..X......
$$ -X.O56X...Q.
$$ -OO.X79.....
$$ -XOOX8......
$$ -.XX13......
$$ -...........
$$ -..2.4......
$$ -...........
$$ -..Q........
$$ -...........
In short, the shape works, but it's so packed with aji that usually, it's better to hold back a little and play more solid.
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 only works when there is any space to develop, so it is advantageous in the early game.)
