A term from the game of Go (Wei Qi/Badouk). An eye is a hollow point inside a group of stones, as shown in this diagram:

......
.ooo..
.o.oo.
.ooo..
....o.
......

The group of stones in the diagram has one eye, the empty point in the center, with eight stones around it. A group needs two eyes (or the guaranteed ability to make two eyes) to live (see my writeup on the eye node for more info). Eye space is an empty area of the board that is surrounded, or almost surrounded by a group of stones, such that the group has potential to make one or more eyes there. In the following diagram, the empty points marked with : instead of . more or less constitute the white (o) group's eye space (the bottom of the diagram is the edge of the board; all other sides are open). The diagram with all the letters is hard to read, so I'll put in two, one without all the letters.

...xx.x... ...xx.x...
...xo..... ...xo.....
.xxoocxx.. .xxoo.xx..
.aob:doox. ..o:::oox.
..e:h::fg. ...::::...

(No stones at a, b, c, d, e, f, g and h, but see below. Note that h is part of the eye space.)

Eye space is a very important concept in life-and-death situations, because the amount (and shape) of a group's eye space will determine whether or not the opponent can kill it. The general advice for trying to kill a group is this: First reduce eye space as much as you can, then play on the vital point (that is, the point that would allow the opponent to split one big eye into two eyes, and thereby live). The reason for this is that playing on what seems to be a vital point doesn't do any good it the opponent can just enlarge his eye space enough to create more ways to make two eyes.

So, how to go about reducing eye space? Well, that's a matter of tactics, and depends on the situation, but in the diagram above, the empty points b and d are eye space for White if he gets to play on the points a and c (respectively) before Black does. If Black plays at a, White will have to connect at b, and if Black plays at c, White will have to play at d to keep Black from getting into the rest of the group's eye space. Of these, the play at a is more important for Black, because if White gets to play there, the point e will also become eye space. Note also that a Black play at f will destroy one point of White's eye space (the point just to the left).

I haven't fully analyzed the situation, so I might be wrong (/msg me if I am), but I suspect that White is dead here if it's Black's turn. Black can play a and then c in sente, then try to reduce White to a dead shape by playing f. Note that Black must then not connect at g, and instead play at h. White can capture the stone at f, but unless there are friendly White stones nearby to connect to, there will be no opportunity to make another eye (the empty point at f would be a false eye).

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