Irrational winding of a torus
In topology, a branch of mathematics, an irrational winding of a torus is a continuous injection of a line into a torus that is used to set up several counterexamples. A related notion is the Kronecker foliation of a torus, a foliation formed by the set of all translates of a given irrational winding.
One way of constructing a torus is as the quotient space of a two-dimensional real vector space by the additive subgroup of integer vectors, with the corresponding projection . Each point in the torus has as its preimage one of the translates of the square lattice in , and factors through a map that takes any point in the plane to a point in the unit square given by the fractional parts of the original point's Cartesian coordinates. Now consider a line in given by the equation y = kx. If the slope k of the line is rational, then it can be represented by a fraction and a corresponding lattice point of . It can be shown that then the projection of this line is a simple closed curve on a torus. If, however, k is irrational, then it will not cross any lattice points except 0, which means that its projection on the torus will not be a closed curve, and the restriction of on this line is injective. Moreover, it can be shown that the image of this restricted projection as a subspace, called the irrational winding of a torus, is dense in the torus.
Irrational windings of a torus may be used to set up counter-examples related to monomorphisms. An irrational winding is an immersed submanifold but not a regular submanifold of the torus, which shows that the image of a manifold under a continuous injection to another manifold is not necessarily a (regular) submanifold. Irrational windings are also examples of the fact that the induced submanifold topology does not have to coincide with the subspace topology of the submanifold  a[›]
Secondly, the torus can be considered as a Lie group , and the line can be considered as . Then it is easy to show that the image of the continuous and analytic group homomorphism is not a Lie subgroup (because it's not closed in the torus – see the closed subgroup theorem) while, of course, it is still a group. It may also be used to show that if a subgroup H of the Lie group G is not closed, the quotient G/H does not need to be a submanifold and might even fail to be a Hausdorff space.
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