Two Eyes Can Die
What is an eye is often confusing to beginners. Beginners may have several misunderstandings regarding eyes. These misunderstandings can lead beginners to believe that it is possible for a group with two eyes to die. Let's explore this:
To be safe from capture, a group of stones needs two irremovable liberties. This is achieved by having two eyes. There are a few, comparatively rare exceptions, such as the situation described by the term two headed dragon.
The crux of the matter is that an eye's status can change if the player allows the opponent to play freely inside on of the one of the eyes of a group.
In this example, Black has played four moves, marked as , inside of one of the White group's eyes. These four moves have removed all but one of the liberties within that eye of the White group. By playing on the last liberty, marked by a red circle in the diagram, Black creates a critical situation. If White ignores Black's moves, Black will be able to capture the White group. If White plays, White can capture the Black stones within the eye of her group. This type of play can continue till there is only one empty point within each eye, making each eye an irremovable liberty.
The White eye, in the corner, ceases to be an eye, however, when it contains a group forming an eye in the belly. Then it turns to a capturing race with different possible outcomes. Here the white group is still alive [1], tangled up in a seki with the black stones in its belly.
Once Black has removed all but one liberty from one of the eyes within the White group, White cannot attack the Black stones. To do so, would leave the White group with only one liberty, in the other eye. Black would play on the one remaining liberty and capture the White group.
This White group is alive: it has one eye in the corner and a big eye space next to it. Technically, this eye space can be occupied with black stones, in a way that captures all white stones:
But such an event is very unlikely to occur, because of alternating play. White would have passed (or tenuki) up to 12 times in a row to get captured.
See also
Notes
[1] That is, alive under normal conditions but not unconditionally alive according to Mr. Benson's definition.