The Wall of Darkness by Arthur C. Clarke (1949) is a piece of Math fiction, usually quoted in semi-academic circles as illustrating the geometry of the Moebius band.
It deals with a universe with a single sun and planet but no other stars. The world supposedly always stayed light, with only a slight change with the sun dipping toward the horizon a bit in winter. Their planet has an inhospitably hot north, a temperate middle, and an extremely cold south. The south is barren, except for an insurmountable wall that stretches across the world at a point so far south that people can barely reach it during the summer, when things warm up.
There is a rumor that seeing what is on the other side of the wall will make a man go mad. But a curious, wealthy guy named Shervane decides he just has to do it anyway. In a massive project that takes more than 7 years, he has a series of platforms built, and he walks up on the wall, making sure his friend will blow everything up if something horrible happens.
Then he walks away from the sun, which is dimming behind him as he walks, and in front of him another sun appears and grows bright. As he approaches the edge of the wall, he sees his friend (whom he left behind) peering up at him.
Then they blow up the platforms, so that no one else can ever try to breach the wall again, saying it was necessary. He imagines in his mind another him blowing up the platform on the other side, but says of course that is impossible, since he is the only man in the world who knows for sure that the WALL HAS ONLY ONE SIDE.
The geometry of the Moebius strip is explained in the story as well, through a Professor of the protagonist.
But of course, as the author must have known well, the Mobius strip is two dimensional and any model of the universe (in his story) has to be three dimensional.
Now a Klein bottle is usually regarded as a higher dimensional equivalent of the Moebius band. While this in clearly a misconception (the Klein "bottle" is a two dimensional surface, just like the Moebius strip, only it seems to enclose a volume unlike the other), it could well be that the model involved here is the "inside" of the Klein bottle, or, equivlently, that the universe has the Klein bottle for its boundary. But it fails me to recognise how exactly the "wall situation" works with a universe in such a shape.
It has been suggested that the wall is built on the neck of the bottle. But a glance at the figure will show that a person who walks "across the top" of such a wall shall see just the region beyond, not the same side. The property of the Klein bottle (analogous to Moebius band having only one side and not two) is that it has only one of the parities left and right, and only one of inside and outside. For our wall to work in the intended way, it has to "go through" the "surface" connecting the false inside and false outside, which does not make any sense.
It has also been commented that the story is trying to sound more impressive than it actually is. But it could be just that the author intended no more than a mood piece focusing on the philosophy of our existence and our search into "the great unknown."
Can anyone point what the proper geometry of this universe is? Which means to recognise which known mathematical shape - a three dimensional surface - models it accurately.
The summary of the story and the comments given above are copied from here.
EDIT: We allow branched 3D surfaces as well when we say 3D surface, which seems necessary in view of the discussion following Kyle Jones' answer.